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/, Paisit Teeraphatsakool/, 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:

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.

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