Gödel’s Incompleteness Theorem: The Universe, Mathematics and God
80 years ago, Kurt Gödel toppled empires of mathematical philosophy with his famous Incompleteness Theorems.
I’m every bit as interested in science, philosophy and engineering as I am in business. Gödel’s theorem has profound implications for every branch of knowledge.
The materialist view prevails in secular circles. Materialism states that the laws of physics and the universe we know are all that is. It sees the universe as a giant machine. It assumes that everything we experience is purely the result of blind cause and effect. It scoffs at the idea that there is any such thing as God or metaphysics.
This view was epitomized by “Logical Positivism” which was espoused by a group known as “The Vienna Circle” in Austria, led by Ludwig Wittgenstein. Logical Positivism says that anything that cannot be experimentally verified or mathematically proven is invalid.
The Logical Positivists were confident that very soon, all the loose ends of mathematics would be nailed down by a single unifying theory. The world would finally fully embrace reason and logic and leave the failures of religion behind.
Kurt Gödel was a member of the Vienna Circle and in 1931 proved that a single unifying theory was impossible. He proved that the goal of the Logical Positivists was unachievable. This was a devastating blow.
Godel’s Incompleteness Theorem says that any system that is complex enough to express mathematics cannot prove, by itself, that everything it says is true. It will always rely on something outside the system that you have to assume is true but cannot prove.
You can then step outside the system and complete your proof, but in order to do that you will now have to invoke something else from the outside. So you keep expanding ever outward, invoking still more things that you cannot prove.
This was very disturbing to mathematicians, because mathematicians hate uncertainty.
Many people have raised the question of whether Gödel’s incompleteness theorem applies to the universe itself. If the universe is mathematical, then yes in fact it does.
Stated in Formal Language:
Gödel’s theorem says: “Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true, but not provable in the theory.”
The Church-Turing thesis says that a physical system can express elementary arithmetic just as a human can, and that the arithmetic of a Turing Machine (computer) is not provable within the system and is likewise subject to incompleteness.
Any physical system subjected to measurement is capable of expressing elementary arithmetic. (In other words, children can do math by counting their fingers, water flowing into a bucket does integration, and physical systems always give the right answer.)
Therefore the universe is capable of expressing elementary arithmetic and like both mathematics itself and a Turing machine, is incomplete.
1. All non-trivial computational systems are incomplete
2. The universe is a non-trivial computational system
3. Therefore the universe is incomplete
Some time ago I posted an article about this: http://www.perrymarshall.com/articles/religion/godels-incompleteness-theorem/ and I was greatly interested in seeing if anyone would be able to poke a hole in my argument. (The article is a much more thorough explanation of Gödel than I am giving you here.)
Nearly everyone agrees that math is incomplete. The idea that the universe is also incomplete apparently makes some people very uncomfortable. If the universe cannot explain itself then there has to be some kind of higher power at work.
The debate essentially comes down to this:
- If the universe is illogical and inconsistent then it is possible for it to be complete.
- If the universe is logical and consistent then it is incomplete.
- If the universe is incomplete, then it depends on something on the outside.
In other words, if the laws of mathematics and logic apply to the universe, then the universe has to have a metaphysical source. Atheism can only be true if the universe is irrational.
(By the way, my experience from conversing with literally thousands of atheists via email and on my various blogs is this: When you get down to the core emotional center of why they don’t believe in God, it’s often because they feel deep down that the universe is irrational. They’re immensely disappointed that the world is full of evil and suffering. Because of this, they reject the idea of God.)
You cannot prove that the universe is mathematical. But belief that the universe is mathematical is the #1 assumption of modern science. Toss out that assumption and the whole philosophical framework of western civilization crumbles.
In the history of science, you will find that belief in a God who created an orderly mathematical universe was one of the foundations of scientific discovery.
If you visit the world’s largest atheist website, Infidels, on the home page you will find the following statement:
“Naturalism is the hypothesis that the natural world is a closed system, which means that nothing that is not part of the natural world affects it.”
If you know Gödel’s theorem, you know that all logical systems must rely on something outside the system. So according to Gödel’s Incompleteness theorem, the Infidels cannot be correct. If the universe is logical, it has an outside cause.
Thus atheism violates the laws of reason and logic.
The Incompleteness of the universe isn’t formal proof that God exists. But… it IS proof that in order to construct a rational, scientific model of the universe, belief in God is not just 100% logical… it’s necessary.
Practically speaking, all knowledge we have about anything is incomplete. There are always some things you’re certain of, some things you’re somewhat sure of, and some things you cannot prove at all. Human knowledge is always enlarging the circle of what is known, but every question that we answer just provokes more questions. In real human experience, the quest to enlarge that circle never stops.
And I would submit to you that this is the essence of faith, as actually practiced by thinking, reasoning people.
Many people assume that religious faith is some mystical imaginary idea that is embraced purely on the basis of emotion or intuition. That it has nothing to do with facts, reason or logic.
This is completely untrue – at least in in the Judeo-Christian tradition. No one is asking you to believe without evidence or rational reason. Belief in God, in Jesus, and even the afterlife is based on historical events, logical propositions, and reasonable arguments.
Science itself originated from theology. Science assumed then, and assumes now, that the universe is rational. That the universe operates according to fixed, discoverable laws. Even science itself is a very practical outworking of faith in the reliability and consistency of the natural order.
The practice of faith is in many ways living out a hypothesis: That if you follow the teachings and embrace the Spirit, you will have an excellent opportunity to experience success in your work and your family. And that you will be rewarded in your search for meaning and pursuit of the deepest questions.
P.S.: If this intrigues you, make sure you read my more extended article, Gödel’s Incompleteness Theorem: The #1 Mathematical Discovery of the 20th Century
About the Author
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