How To Mathematically Prove The Existence of God (Or Not)

Remember to carry the 1, today we're proving the existence of God. Image credit: Freeda Michaux/Shutterstock.com, Paisit Teeraphatsakool/Shutterstock.com, IFLScience

OK, so maybe the existence (or otherwise) of an all-powerful benevolent creator that transcends the bounds of mortal understanding isn’t something that can be proven mathematically. But that doesn’t mean people haven’t tried.

From Blaise Pascal, the 17th-century mathematician who bet on belief, to the modern age of computers, history is full of people who brought an equation to a bible fight. Here are a few of the most famous examples.

Blaise Pascal Bets on God

Blaise Pascal didn’t really intend for his “wager” to be proof that God existed – he really just wanted to convince people to act like "He" did.

Pascal’s biggest contribution to the world of math was probably (haha) the development of probability, and this was what his argument was based on. Either God exists, he said, or He doesn’t. If God exists, and you believe in Him, you get to go to heaven forever, which is a pretty good deal all things considered. If you don’t believe, though, you’re damned to hell, which is at best a suboptimal outcome.

On the other hand, if God doesn’t exist, then whether or not you believe in Him really doesn’t matter, continued Pascal. At most, you get to feel a little smug. So overall, he said, the better option is to believe in God – or at least to live as if you do.

“Pascal’s wager”, as it’s known, was dismissed by atheists for being “not really proof” and by theists for being “not really belief”, but it’s nevertheless an intriguing argument – and it’s currently seeing a new life in the age of face masks and climate change.

Pascal's wager. Image Credit: WikiHow

Leonhard Euler Bamboozles the Unbeliever

In the late 18th century, the court of Catherine the Great was the place to be if you were an Enlightenment thinker. And it was there that, as legend has it, the atheist philosopher and writer Denis Diderot was left dumbstruck by a mathematical proof of God put forward by Leonhard Euler.

“Sir,” he announced to Diderot in front of the court, “(a+b^n)/n=x, hence God exists; reply!”

As you probably suspect, the statement is nonsense, but if the legend is to believed (and like all the best legends, it probably isn’t), Diderot knew nothing about math, and immediately left Russia ashamed at being so publicly “proven” wrong.

KURT Gödel Brings Back a Classic

We tend to assume that the idea of openly debating the existence of God is a modern phenomenon, but it's not true. The first known example of what philosophers call an "ontological" argument for the existence of God was published nearly a thousand years ago, during a period of history that isn't but could legitimately be known as the "so religious that we will happily destroy medical textbooks to put prayers in there instead" era. It came from a Benedictine monk called Anselm (just Anselm, like Cher or Madonna) who was later promoted to Archbishop of Canterbury and then later further promoted to Saint, so clearly the church thought the argument was pretty good.

It went basically like this: by definition, God is the greatest thing that can possibly exist.

Now, God either exists, or doesn't exist. Let's assume for the moment that He doesn't.

But if God can possibly exist, but doesn't, then it's possible to think of a greater being, to wit, something that is the same as this "God" but also exists.

So we're now faced with the idea of the existence of something greater than the greatest thing that can possibly exist. This, Anselm argued, is clearly absurd, and therefore our assumption that God doesn't exist must be false.

Now it might be a bit medieval and logically shaky, but this is a pretty good example of what modern-day mathematicians call a proof by contradiction. You assume something is true, show that that assumption logically leads to complete nonsense, and conclude that the assumption must therefore be false. It's a mainstay of the mathematician's arsenal, so it makes sense that this was the idea that legendary mathematician Kurt Gödel revamped for his own ontological argument nearly 900 years later.

If you're hoping that being written in the 20th century will make the "proof" easier to read, though, I've bad news: Gödel was a logician, and he's responsible for some of the most abstract mathematics it's possible to conceive of. So when Gödel wrote about God, it looked like this:

content-1623174079-godel.jpg
Hope that clears things up for you.

Let's translate that into English.

OK, “Ax. 1” means “Axiom 1”. Axioms are like the atoms of math: little truths so fundamental that we either can’t prove them, or we don’t want to because they’re so self-evident. So, for example, “x=x” is an axiom: we can’t prove it, it just is (“1+1=2”, however, we do need to prove).

Gödel uses his axioms to set out his idea of a “positive property”. First, he says that if the property φ is “positive”, and also the property φ implies the property ψ, then the property ψ is also positive. Next, in axiom two, he tells us that either a property is positive, or its negation is positive, but not both.

So, to take a simple example, let’s say φ is the property “being blue in color”. Axiom 1 says that if being blue in color is a positive property, then being not red is also a positive property, because being blue by definition means you are not red. Axiom 2 then says that being red must not be a positive property, because its negation is positive.

Now we get onto Gödel’s first theorem, which is the following: if φ is a positive property, then it’s possible that something, somewhere, exists that has this property. Seems fair enough.

Next, a juicy definition: Godlikeness. Something is godlike, Gödel says, if it has every possible positive property. What’s more, he says, being godlike is itself a positive property, which we can’t argue with, because he’s made it an axiom.

Gödel now has what he needs to lay out a big theorem: here, he says that since something, somewhere, exists that has every possible positive property, and being godlike is a positive property, that means that something, somewhere, exists which is godlike.

Gödel’s next step is to show that this godlike thing, which we may as well call God, therefore exists everywhere, and he does this by introducing the idea of “essences”.

The first line here defines an “essence” of an object x as a property of x which necessarily implies all other properties of x. So for instance, we might say that “puppyhood” is an essence, because if we know something is a puppy, we automatically also know it’s cute and fluffy and a very good boy or girl.

Gödel then says it’s an axiom that a property being positive somewhere means it’s positive everywhere – arguably true for puppies but perhaps less so for more morally nuanced things like veganism or mimes.

Theorem 3 says that if something is godlike, then that is its defining essence. That pretty much makes sense: being godlike is defined in terms of every possible other property an object can have, so saying something is godlike does indeed tell us everything else we could possibly want to know about it.

He then defines the concept of “necessary existence”. An object exists, he says, if something exists somewhere which has its essential property. So (you’ll be relieved to know) we can logically say that puppies exist, because there are definitely things in the world which have the property of puppyhood (for instance: puppies).

Amen.

And now for the payoff. Existence, Gödel states, is a positive property. But God has every positive property. And what’s more, something which is positive here is positive everywhere. QED, God exists, says Gödel.

Now, you may have noticed that there are a few problems with this “proof”, and we’ve already touched on the main one: Gödel simply never gave any reasoning for any of his axioms. Mathematically, this means that there are zero reasons to believe his conclusions are true. Philosophically, it means there are zero reasons to believe anything is true. Gödel was a genius, and may have convinced himself of the existence of God, but he certainly didn’t prove it.

Deus Ex Machina

So perhaps a mathematical proof of God is simply too hard for humans to create – but what if we could get a machine to do it for us?

In 2013, two computer scientists made headlines when they uploaded a paper to the preprint server arXiv titled “Formalization, Mechanization and Automation of Gödel's Proof of God's Existence”. They showed – on a MacBook, no less – that Gödel’s conclusion was correct.

At least, assuming his axioms were correct. The truth was that the scientists hadn’t set out to make a theological statement, but a scientific one: all they wanted to do was show off their algorithm.

The debugging of Adam. Image Credit: Blue Planet Studio/Shutterstock.com

QED?

While many people have attempted over the years to use math to prove the existence of God, nobody has succeeded yet – and it’s unlikely that anybody ever will.

Of course, for many believers, that’s the point. But if you still want to try proving it, there are worse ways to start than by studying math.


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