Gödel’s Incompleteness: The #1 Mathematical Breakthrough of the 20th Century
Every now and again I come across a fantastic article the warrants posting here; I recently came across one CosmicFingerprints.com which, I thought, was pretty insightful. Be edified.
“Faith and Reason are not enemies. In fact, the exact opposite is true! One is absolutely necessary for the other to exist. All reasoning ultimately traces back to faith in something that you cannot prove.”
In 1931, the young mathematician Kurt Gödel made a landmark discovery, as powerful as anything Albert Einstein developed.
In one salvo, he completely demolished an entire class of scientific theories.
Gödel’s discovery not only applies to mathematics but literally all branches of science, logic and human knowledge. It has earth-shattering implications.
Oddly, few people know anything about it.
Allow me to tell you the story.
Mathematicians love proofs. They were hot and bothered for centuries, because they were unable to PROVE some of the things they knew were true.
So for example if you studied high school Geometry, you’ve done the exercises where you prove all kinds of things about triangles based on a set of theorems.
That high school geometry book is built on Euclid’s five postulates. Everyone knows the postulates are true, but in 2500 years nobody’s figured out a way to prove them.
Yes, it does seem perfectly “obvious” that a line can be extended infinitely in both directions, but no one has been able to PROVE that. We can only demonstrate that Euclid’s postulates are a reasonable, and in fact necessary, set of 5 assumptions.
Towering mathematical geniuses were frustrated for 2000+ years because they couldn’t prove all their theorems. There were so many things that were “obviously true,” but nobody could find a way to prove them.
In the early 1900’s, however, a tremendous wave of optimism swept through mathematical circles. The most brilliant mathematicians in the world (like Bertrand Russell, David Hilbert and Ludwig Wittgenstein) became convinced that they were rapidly closing in on a final synthesis.
A unifying “Theory of Everything” that would finally nail down all the loose ends. Mathematics would be complete, bulletproof, airtight, triumphant.
In 1931 this young Austrian mathematician, Kurt Gödel, published a paper that once and for all PROVED that a single Theory Of Everything is actually impossible. He proved they would never prove everything. (Yeah I know, it sounds a little odd, doesn’t it?)
Gödel’s discovery was called “The Incompleteness Theorem.”
If you’ll give me just a few minutes, I’ll explain what it says, how Gödel proved it, and what it means – in plain, simple English that anyone can understand.
Gödel’s Incompleteness Theorem says:
“Anything you can draw a circle around cannot explain itself without referring to something outside the circle – something you have to assume but cannot prove.”
You can draw a circle around all of the concepts in your high school geometry book. But they’re all built on Euclid’s 5 postulates which we know are true but cannot be proven. Those 5 postulates are outside the book, outside the circle.
|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
You can draw a circle around a bicycle. But the existence of that bicycle relies on a factory that is outside that circle. The bicycle cannot explain itself.
You can draw the circle around a bicycle factory. But that factory likewise relies on other things outside the factory.
Gödel proved that there are ALWAYS more things that are true than you can prove. Any system of logic or numbers that mathematicians ever came up with will always rest on at least a few unprovable assumptions.
Gödel’s Incompleteness Theorem applies not just to math, but to everything that is subject to the laws of logic. Everything that you can count or calculate. Incompleteness is true in math; it’s equally true in science or language and philosophy.
Gödel created his proof by starting with “The Liar’s Paradox” — which is the statement
“I am lying.”
“I am lying” is self-contradictory, since if it’s true, I’m not a liar, and it’s false; and if it’s false, I am a liar, so it’s true.
Gödel, in one of the most ingenious moves in the history of math, converted this Liar’s Paradox into a mathematical formula. He proved that no statement can prove its own truth.
You always need an outside reference point.
The Incompleteness Theorem was a devastating blow to the “positivists” of the time. They insisted that literally anything you could not measure or prove was nonsense. He showed that their positivism was nonsense.
Gödel proved his theorem in black and white and nobody could argue with his logic. Yet some of his fellow mathematicians went to their graves in denial, believing that somehow or another Gödel must surely be wrong.
He wasn’t wrong. It was really true. There are more things that are true than you can prove.
A “theory of everything” – whether in math, or physics, or philosophy – will never be found. Because it is mathematically impossible.
OK, so what does this really mean? Why is this super-important, and not just an interesting geek factoid?
Here’s what it means:
- Faith and Reason are not enemies. In fact, the exact opposite is true! One is absolutely necessary for the other to exist. All reasoning ultimately traces back to faith in something that you cannot prove.
- All closed systems depend on something outside the system.
- You can always draw a bigger circle but there will still be something outside the circle.
Reasoning inward from a larger circle to a smaller circle (from “all things” to “some things”) is deductive reasoning.
Example of a deductive reasoning:
1. All men are mortal
2. Socrates is a man
3. Therefore Socrates is mortal
Reasoning outward from a smaller circle to a larger circle (from “some things” to “all things”) is inductive reasoning.
Examples of inductive reasoning:
1. All the men I know are mortal
2. Therefore all men are mortal
1. When I let go of objects, they fall
2. Therefore there is a law of gravity that governs all falling objects
Notice than when you move from the smaller circle to the larger circle, you have to make assumptions that you cannot 100% prove.
For example you cannot PROVE gravity will always be consistent at all times. You can only observe that it’s consistently true every time.
Nearly all scientific laws are based on inductive reasoning. All of science rests on an assumption that the universe is orderly, logical and mathematical based on fixed discoverable laws.
You cannot PROVE this. (You can’t prove that the sun will come up tomorrow morning either.) You literally have to take it on faith. In fact most people don’t know that outside the science circle is a philosophy circle. Science is based on philosophical assumptions that you cannot scientifically prove. Actually, the scientific method cannot prove, it can only infer.
(Science originally came from the idea that God made an orderly universe which obeys fixed, discoverable laws – and because of those laws, He would not have to constantly tinker with it in order for it to operate.)
Now please consider what happens when we draw the biggest circle possibly can – around the whole universe. (If there are multiple universes, we’re drawing a circle around all of them too):
- There has to be something outside that circle. Something which we have to assume but cannot prove
- The universe as we know it is finite – finite matter, finite energy, finite space and 13.8 billion years time
- The universe (all matter, energy, space and time) cannot explain itself
- Whatever is outside the biggest circle is boundless. So by definition it is not possible to draw a circle around it.
- If we draw a circle around all matter, energy, space and time and apply Gödel’s theorem, then we know what is outside that circle is not matter, is not energy, is not space and is not time. Because all the matter and energy are inside the circle. It’s immaterial.
- Whatever is outside the biggest circle is not a system – i.e. is not an assemblage of parts. Otherwise we could draw a circle around them. The thing outside the biggest circle is indivisible.
- Whatever is outside the biggest circle is an uncaused cause, because you can always draw a circle around an effect.
We can apply the same inductive reasoning to the origin of information:
- In the history of the universe we also see the introduction of information, some 3.8 billion years ago. It came in the form of the Genetic code, which is symbolic and immaterial.
- The information had to come from the outside, since information is not known to be an inherent property of matter, energy, space or time.
- All codes we know the origin of are designed by conscious beings.
- Therefore whatever is outside the largest circle is a conscious being.
When we add information to the equation, we conclude that not only is the thing outside the biggest circle infinite and immaterial, it is also self-aware.
Isn’t it interesting how all these conclusions sound suspiciously similar to how theologians have described God for thousands of years?
Maybe that’s why it’s hardly surprising that 80-90% of the people in the world believe in some concept of God. Yes, it’s intuitive to most folks. But Gödel’s theorem indicates it’s also supremely logical. In fact it’s the only position one can take and stay in the realm of reason and logic.
The person who proudly proclaims, “You’re a man of faith, but I’m a man of science” doesn’t understand the roots of science or the nature of knowledge!
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 all systems rely on something outside the system. So according to Gödel’s Incompleteness theorem, the folks at Infidels cannot be correct. Because the universe is a system, it has to have an outside cause.
Therefore Atheism violates the laws mathematics.
The Incompleteness of the universe isn’t proof that God exists. But… it IS proof that in order to construct a consistent model of the universe, belief in God is not just 100% logical… it’s necessary.
Euclid’s 5 postulates aren’t formally provable and God is not formally provable either. But… just as you cannot build a coherent system of geometry without Euclid’s 5 postulates, neither can you build a coherent description of the universe without a First Cause and a Source of order.
Thus faith and science are not enemies, but allies. They are two sides of the same coin. It had been true for hundreds of years, but in 1931 this skinny young Austrian mathematician named Kurt Gödel proved it.
No time in the history of mankind has faith in God been more reasonable, more logical, or more thoroughly supported by rational thought, science and mathematics.
“Math is the language God wrote the universe in.” –Galileo Galile, 1623
“Incompleteness: The Proof and Paradox of Kurt Gödel” by Rebecca Goldstein – fantastic biography and a great read
A collection of quotes and notes about Gödel’s proof from Miskatonic University Press
Formal description of Gödel’s Incompleteness Theorem and links to his original papers on Wikipedia
Science vs. Faith on CoffeehouseTheology.com
Originally published on cosmicfingerprints.com and can be found here.