I just started reading Ed Feser’s Five Proofs for the Existence of God. It is what I expected it to be, and I’m going to critique each of the five arguments because I don’t think they are very good arguments, certainly not rising to the level of ‘proof.’

### Argument vs. Proof

Well, what is a proof? A proof is a demonstration of the connection between a logical or mathematical statement on one hand, and a set of axioms, on the other hand, using only an accepted set of logical, inferential operations.

Proofs of God’s existence usually trade on sloppy definitions, imprecise arguments, and equivocation. All of this is easy in arguments made in human language, but it falls apart when you try to recast the same arguments in strictly logical terms.

But more fundamentally, is God a theorem or an axiom?

Axioms are the ideas we have to take for granted. From them, we can derive theorems, which depend completely on our choice of axioms. And it is a choice, we can use any set of axioms we want. For example, in classical geometry, there is an axiom known as the Parallel Postulate. One way to state the Parallel Postulate is that parallel lines never meet even if extended to infinity. Another way to put it, that avoids the concept of infinity, is that the number of degrees in the sum of the interior angles of a triangle is exactly 180.

Because the Parallel Postulate was cumbersome to state, considerable effort was expended to show that it wasn’t an axiom – that it could be derived from the other axioms of Euclidean geometry. However, mathematicians in the early 19th century showed that it really was an axiom. By changing the axiom to versions where the interior angles added up to more or less than 180 degrees, entirely new geometries were invented. In particular, spherical geometry uses the greater than 180 assumption, and is necessary to do accurate surveying on the surface of the Earth.

The point here is that what you really are expressing faith in is your axioms, not your theorems. So if you really can ‘prove’ God exists under some system of axioms, you aren’t expressing faith in God, you’re expressing faith in something else.

And BTW, if your set of axioms contains a contradiction, you can prove anything.

So when critiquing these arguments I’ll focus much more on critiquing the axioms, rather than the derived ideas.

### Infinity

One of those key assumptions is that infinite sequences can’t be real. This idea appears several times in arguments that there must be a First Mover, a First Cause because an infinite sequence just can’t be accepted as a possibility. If we eliminate an infinite regress, there must be a First, right? Not so fast.

Imagine you are a tiny, point-like creature that lives on a circle. You want to back up to where you were before. And again, And again. Are you ever going to finish backing up? Obviously not, because you live on a circle, you will eventually back up completely around the circle and keep going around and around. While the circle is finite, it is also unbounded.

“Finite, but unbounded” is just one way of avoiding the need for a First. Another is to point out that we do experience ‘actual infinites’ all the time. Ancient philosophers may have had trouble with this idea but subsequent generations have given us better tools for thinking.

If you took calculus in school, you know it works by taking a limit as something goes to infinity. Taylor series and Fourier series both work by summing an infinite series of terms, which can converge to a finite, useful number. For example, a square wave is the sum of an infinite number of sine waves.

A deeper, physical example is the Feynman integral way of understanding quantum mechanics. In this understanding, particles don’t move along a single straight line, they move along an infinite number of paths simultaneously. The sum of all these motions is the appearance of a straight line in simple cases, but can explain more complex cases.

We can also consider the path of a photon as it moves outward. A ray, geometrically, is infinite in one direction. There was a time in the early history of the universe when it became optically transparent for the first time. At the moment a huge amount of the mass/energy of the universe which was photons escaped outwards at the speed of light. Nothing will ever absorb these photons. They will speed onwards forever.

You might object the future is not as real as the past. While it certainly is not as well known, the broad outline is well enough known to say that if the universe expands infinitely, these photons will also last forever. Photons don’t decay into other particles.

### Infinite theorems, unprovably true

Mathematical development since the end of the 19th century and into the mid-20th century led to a confluence of the ideas of proof and infinity. We discovered that any logical system complex enough to support the development of arithmetic also led to an odd circumstance. There would be theorems in the system that were true, but unprovable within that system. If you promoted them from theorems to axioms there would still be more theorems that were true but unprovable. For more on this fascinating topic, look into *Gödel’s incompleteness* theorems.

Now any explanation of physics is going to satisfy this condition – it is a logical system that is strong enough to include arithmetic. Therefore we can expect that there are going to be corners of physics that also fall into the category of true, but unprovable.

And of course there is no way to know if any particular idea is unprovable or not!

### Why?

Some readers might ask why I am getting into this subject here on Rational Exuberance. I’ve previously taken on books like this on another blog. But I want to do this here because there is a connection between this kind of thinking and the political issues that are more commonly discussed here.