Gödel's Incompleteness Theorem:
The #1 Mathematical Discovery of the 20th Century
In 1931, Kurt Gödel delivered a devastating blow to the mathematicians of his time
In 1931, the young mathematician Kurt Gödel made a landmark discovery, as powerful as anything Albert Einstein developed.
Gödel's discovery not only applied to mathematics but literally all branches of science, logic and human knowledge. It has truly 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 list 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 reasonable that a line can be extended infinitely in both directions, but no one has been able to PROVE that. We can only demonstrate that they 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 many things that were "obviously" true but nobody could figure out a way to prove them.
In the early 1900's, however, a tremendous sense of optimism began to grow in mathematical circles. The most brilliant mathematicians in the world (like Bertrand Russell, David Hilbert and Ludwig Wittgenstein) were 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.
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 discovered 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 are clearly true but cannot be proven. Those 5 postulates are outside the book, outside the circle.
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.
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. Incompleteness is true in math; it's equally true in science or language or philosophy.
And: If the universe is mathematical and logical, Incompleteness also applies to the universe.
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.
So Gödel, in one of the most ingenious moves in the history of math, converted the Liar's Paradox into a mathematical formula. He proved that any statement requires an external observer.
No statement alone can completely prove itself true.
His Incompleteness Theorem was a devastating blow to the "positivism" of the time. 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 than are true than you can prove.
A "theory of everything" – whether in math, or physics, or philosophy – will never be found. Because it is 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 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 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 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. You cannot prove that the universe is rational. You can only observe that mathematical formulas like E=MC^2 do seem to perfectly describe what the universe does.
Nearly all scientific laws are based on inductive reasoning. These laws rest on an assumption that the universe is logical and 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.)
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 13.7 billion years time
- The universe (all matter, energy, space and time) cannot explain itself
- Whatever is outside the biggest circle is boundless. 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. 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 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.
In other words 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 conscious.
Isn't it interesting how all these things sound suspiciously similar to how theologians have described God for thousands of years?
So 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!
Interesting aside…
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 systems must rely on something outside the system. So according to Gödel's Incompleteness theorem, the Infidels cannot be correct. Because the universe is a logical system, it has an outside cause.
Thus atheism violates the laws of reason and logic.
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. It's 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 reason, logic, science and mathematics.
Perry Marshall
"Math is the language God wrote the universe in."
Further reading:
"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 on Wikipedia
Science vs. Faith on CoffeehouseTheology.com
Information Theory: "If you can read this, I can prove God exists"
Trackback URI
http://www.perrymarshall.com/articles/religion/godels-incompleteness-theorem/trackback/
Perry, it still looks to me as though several assumptions are being made. For instance, this topic is called "Godel's Incompleteness Theorem".
It's not Godel's Incompleteness LAW. It's a theorem. So, even though it describes what we might not know about something else….the fact that it's a theorem says that we can't 100% know for sure that Godel's perspective is right.
Take the first line: "Draw a circle around anything and you must go outside that circle to explain it." That is certainly true with bicycles and gravity…..but just as you mentioned that the sun might NOT come up tomorrow (and that is taken on faith), it's equally possible that Godel's theorem doesn't apply to everything.
For instance, you stated that — if we draw a circle around the Universe (or all the possible Universes) — "there has to be something outside that circle. Something which we have to assume but cannot prove"
That's not necessarily true. We might not even need to assume it. It's quite possible that drawing a circle around everything all the known universes would include a perfect explanation and provide a Single Unifying Law.
"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". — Again, this is assumptive. How do you know that it cannot explain itself? This would only be a true statement if we actually had all the knowledge about the Universe (which we don't).
"Whatever is outside the biggest circle is boundless." There is nothing outside the circle. By any standard definition of "universe" or "universes", if it exists, you've drawn a circle around it. It's bad math to say "Draw a circle around all the known universes" and then to refer to something outside the universe. If there were something outside the circle, you would have drawn the circle to include it!
We can go on and on…but (with respect) this doesn't seem to be a logical argument. It seems to be an argument based on certain presuppositions. And I'm not saying that's good, bad, right or wrong….just that this argument has too many assumptions to conclude that there's no possibility of a Unifying Theory.
Tony,
If Gödel can be shown to be wrong, then you have something to stand on.
80 years have gone by and no one has demonstrated a flaw in Gödel's logic.
If Gödel's Incompleteness theorem is true, then everything in my article directly and logically follows. Specifically, if we draw a circle around all matter, energy, space and time and label it "Universe" then according to Incompleteness, there has to be something outside the universe that caused it.
If Gödel's argument is logical, then my argument is also logical.
You are invited to demonstrate a flaw in Gödel's logic if you can. And I will certainly be happy to post that argument here on my blog.
Perry Marshall
"I am lying." Before I can lie I have to exist, and not only exist but conceive of myself. To conceive of myself I must split "me" as the subject that knows whether I am lying or not, from the "me" as an "object" that is either lying or not lying. That split is an artifice of the mind – reality is not split. It just "is". That reality is not outside the circle of "me" as the subject, or "me" as the object that may or may not be lying. The perception of intelligence is merely the illusion of "self" as separate from reality. It is a trick of the mind.
All perception is the splitting of reality into a subject and an object.
Maybe you want to define god as the indivisible sum of all that is real? A bit superfluous, but it has it's uses.
Steve,
A sum is divisible, so we can't define God that way. We do have to define God as indivisible. God is one.
I like your explanation of perception. If we define God as self-aware, then we automatically invoke a splitting like what you refer to. Which is where the Trinity comes from. Self, expression of self, and self-understanding (Father / Son = WORD / Holy Spirit).
If you like my explanation of perception god cannot be self aware. I prefer god is just reality. This is my justification for my atheism.
Full awareness is to be one with reality, one with god, and the self disappears. As a self we use science to explore objective reality and spiritual practice to explore the subjective reality. Both are lies, but by knowing the lies the truth is revealed. You cannot know it and survive, there is no split. You cannot know reality as there is no knower.
IF there is no knower, then how do you KNOW that "you cannot know reality" and how do you KNOW that "there is no knower?"
Perry, I'm not making this an issue of "right" or "wrong". It's simply a matter of workability.
Draw a circle around all matter, energy, space and time and label it "Universe" and you cannot definitively say that it is dependent on something else outside it. You can speculate that it might be….but there is where the line is drawn between theorem and law.
Simply put, we don't have enough information to say emphatically that Godel's theorem is true. That's why it's a theorem. We can point to multiple ways that it IS true. But, it still makes its own assumptions in matters that cannot be observably demonstrated.
Thanks for the dialogue!
Tony
A theorem is not the same as a theory. Gödel wrote a formal proof of his theorem. You can read his proof here:
On formally undecidable propositions of Principia Mathematica and related systems
This is why we have enough information to say emphatically that Gödel's theorem is true. To say otherwise is to reject the very process of mathematics, proof and logic.
I agree with your logic, Perry (and Godel's), thanks for sharing.
Any assumptions that are made when discussing where the universe came from seem to me to be related to the law of causality…anything that begins to exist has a cause. I think that's a safe assumption.
Philosopher William Lane Craig puts it like this:
Premise 1): Anything that begins to exist has a cause.
Premise 2): The universe began to exist.
Conclusion: The Universe had a cause.
1) The ability of things popping into existence from nothing without a cause is not worth arguing.
2) Both science and philosophy support the idea that the universe had a beginning.
Therefore…it must have had a cause.
It doesn't specifically point to the Christian God that I believe in, I have other reasons for that belief.
However, based on Godel and a finite universe I don't think one can argue that the idea of a god of some sort is illogical.
What caused God? It's not a coherent question. By definition God is eternal…uncaused…was always there, unlike the finite universe.
So "What caused God?" becomes "What caused the un-caused being?" Doesn't make sense.
Always interested in this kind of discussion, though don't usually have so much to say…thanks for posting!
Rod MacKenzie
Rod, you're right. And if Gödel's theorem applies to a rock then it applies equally to a planet and equally to all planets and to the whole universe. There's nothing about zooming in or zooming out that suddenly changes everything.
Perry,
I checked out your talk on Information Theory: "If you can read this, I can prove God exists" on the Cosmic Fingerprints site and found it excellent.
I would like to offer a link to this talk in a Blog I'm working on re: the rationality of the Christian worldview…I had already planned my next post around the same topic.
Do you mind? If not, should I just send readers to the Cosmic Fingerprints site, or do you have this talk available on one of your own sites that you'd prefer I link to?
Thanks,
Rod
Rod,
Use it in any way you want. Also there's a link "Origin of Life Video" which is similar – that may be useful too. I salute your efforts – nice site you have!
Perry
Thanks Perry,
I look forward to reading more interesting stuff from you, and I'll definitely be linking to Cosmic Fingerprints from my Rational Faith site.
Rod
It's also interesting that so many that have posted here can point to the universe and say that since it exists, it must have a cause, and if there is a cause then a god of some form must exist in order to cause it.
If it is accepted as true that nothing can exist without a creator, then God CANNOT exist without being caused by something else.
If you claim that God exists, and can do so without a cause since that is part of the definition of what it means to be God, then it MUST follow that other things (such as the universe) could do so without a creator as well.
The reason the universe cannot be uncaused is because of entropy. If it were infinitely old, there would be no available energy remaining. You can't burn a candle twice. Modern cosmology has established that the universe is approximately 13.7 billion years old and began in a single point expanding with the big bang.
Gödel's theorem shows that the cause of the universe has to be outside the universe, since the universe is by necessity incomplete.
The single direction of entropy and our perception of an "arrow of time" are closely bound together (we remember the past and not the future, and perceive everything as inexorably rolling from the past into the future, and at the same time from a low entropy state to a high entropy state) but it is conceivable that we may simply be in a region of a larger universe, where chance fluctuation created a (temporary) condition of low entropy, that is now returning to a 'baseline' level of entropy over billions of years.
In this wider universe, without the steady progression of entropy there might equally be little in the way of an orderly flow of time, matter interacting in ways that are very hard to imagine intuitively (I don't claim to be able to understand what that would be like, which I'm aware must sound frustratingly similar to the "it's a mystery" so often put forward as a nonexplanation by religion, but the model is workable, and it would be a way to explain our universe without appeal to a truly unknowable supernatural force.
The 'outer' universe would be an inhospitable place for our kind of life, but the idea would have some accord with the multiverse idea; relatively distinct universes expanding into their own pockets or bubbles of space.
I'm not saying that's how it happened, or that it's how I believe it happened; my stance would be to file it under "maybe plausible, pending further evidence".
Matt,
I think it's significant that in order to avoid the obvious consequences of entropy you have to invoke an undetectable universe in which entropy works differently than it works here.
I'd have a hard time calling such a theory parsimonious. Or scientific.
Are you unconditionally committed to atheism as a worldview or are you willing to follow the evidence where it leads?
On what basis do you assert that a supernatural force is unknowable?
Very interesting stuff.
This is a bit off topic but I find myself saying, "Yep; something must have created the universe", but then the next thing that pops into mind is "so what"? Why do people go to such extraordinary lengths to 'prove' it to others (like you do Perry
? Even going as far as killing people.
How does that faith help us or impact our lives? Obviously the answer to that depends on each person's concept of what the creator is, but to me it doesn't help and shouldn't make a bit of difference to how we behave.
Alasdair,
There is a huge battle in the marketplace of ideas regarding the existence of God. Look at how many books on this topic are bestsellers during the last few years. Yes, I have gone to great lengths…. Even to the point of writing an 1800 word summary of Gödel's theorem :^>
Actually I have done much more than just that, which you can see if you visit my other websites http://www.coffeehousetheology.com and http://www.cosmicfingerprints.com.
I believe that no civilization rises higher than its idea of God. A lot of people consider faith to be a private personal thing which others should not be bothered with; but I think that idea is false. Our faith, whatever it may be, greatly affects what we do.
The US Declaration of Independence says, "We hold these things to be self-evident, that all men are endowed with certain inalienable rights, including life, liberty and the pursuit of happiness."
The very idea of human rights is a faith statement. It is not scientifically provable. It can't be derived from Darwinism. It comes from a belief not only in God but God's relation to mankind.
These things matter a lot.
A counterexample would be the abuses of communism in the 20th century. Is it merely a coincidence that the governments who killed more than 100 million people just happened to be officially atheistic? Atheist regimes killed more people in one century than religious wars killed in all centuries put together. Could that really be just a happy accident?
A few articles I think might provoke more thinking on this:
http://www.perrymarshall.com/merry-christmas-2008/
http://www.cosmicfingerprints.com/blog/faq
To the average guy on the street it seems like people are good or bad simply because they're either good people or bad people and religion has nothing to do with it. But I would challenge you to press further: WHY should I believe in human rights? WHY should we take care of sick people and handicapped people? WHY should we try to eradicate poverty? What happens when you drill down to bedrock on those questions? What do you find?
Perry
Perry says:
'You are invited to demonstrate a flaw in Gödel's logic if you can. And I will certainly be happy to post that argument here on my blog.'
Well, it's not that hard to point out the flaw in Godel's argument. It isn't something that can be put into a few lines, but you can see it here:
http://jamesrmeyer.com/godel_flaw.html
and in simplified form, together with a simplified explanation of Godel's proof here:
http://jamesrmeyer.com/ffgit/GodelSimplified0.html
James,
Q: Are you only seeking to show a flaw in Gödel's original paper, or are you seeking to broadly show that the Incompleteness theorem in its various current forms is fundamentally incorrect?
Perry
James,
I am not sufficiently trained in formal mathematics to decide, myself, whether you are right or wrong. Perhaps I could with time but that would require a very significant time commitment from me. Readers can decide for themselves.
What I will say is that based on reading the online debates about this for a couple of hours, I'm not persuaded that you're correct. I sifted through these threads:
http://mathforum.org/kb/message.jspa?messageID=6300764&tstart=0
and
http://groups.google.com/group/sci.logic/browse_frm/thread/3fd9e2fe7b924c74/270a6b8f731207cf?hl=en&pli=1
and
http://r6.ca/Goedel/FFGITReview.html
and
http://www.jamesrmeyer.com/ffgit/review-oconnor.html
I believe everyone is being fair with you. Also I believe that if you are correct about this, you can get this published. There is surely a mathematics journal that would be willing to risk opposing the Gödel doctrine. Such a thing would be a newsworthy event.
My own judgment is based on several things:
1) The conversations I see here are generally not vitriolic. There is real discussion happening here. I know what knuckleheads sound like when they're in denial of losing an argument, and these people are neither knuckleheads, nor are they losing the argument.
2) This is not a shades of gray issue with dozens of difficult-to-quantify factors, like arguing about, say, the myriad causes of global warming. This is math and logic.
3) Gödel's theorem has been scrutinized and obsessed over for decades. The Logical Positivists in particular had enormous motivations to disprove Incompleteness when Gödel first published his paper; yet they could not. I find it difficult to believe that Wittgenstein and Bertrand Russell and even Hilbert would have been unable to find this flaw 50+ years ago if it were real. Not to mention thousands of others who have followed.
Again, readers can judge for themselves. In my opinion there is a slim possibility that the mathematical profession has been wrong about Gödel for 80 years but it's not a horse I'm willing to bet on.
If you are right then I salute you and in any case encourage you to consider getting your findings published.
I am a mathematician, and I can tell you for sure that you have a basic misunderstanding of what the theory of everything is; the theory of everything is a grand unifying field theory. It has nothing to do with having a system that is entirely self contained, but rather a system that can describe the fundamental forces simply. Your assumption that a conscious being must be the external observer seems kind of ridiculous to me, and is a giant jump. You might as well be Descartes with his cogito ergo sum, ergo deus. What a ridiculous notion.
A system that describes the fundamental forces simply is still a system, and still subject to Gödel's theorem.
You're welcome to challenge any specific statement I have made. I maintain that everything I have said here is 100% logical and that the conclusions follow naturally from the premises.
Perry,
I thoroughly enjoyed your article, and the mental, logical, and theological exercise it plays in. There's nothing like hangin' out with smart people to make you smarter.
So frankly, with all the smart people you attract, I'm surprised no one's mentioned this…
Just as "I am lying" is a paradox, so is Godel's theorem. All we need to do is draw a circle around all the things that Godel's theorem applies to. Outside of that circle must be all the things that the theorem doesn't apply to. Therefore, Godel's theorem does not apply to all the things that Godel's theorem applies to.
Oh! (smoke coming from my ears) My head hurts!
Is there a flaw in this application of Godel's theorem?
-Mark
p.s. All my best wishes for a healthy, happy, giving and prosperous new year, Perry.
Mark,
Gödel's theorem applies to all systems, statements, objects and propositions.
The thing outside the biggest circle is not a system or statement or object or proposition. It's real not imaginary; it's axiomatic; it's conscious; it's boundless and immaterial.
Which is to say, Gödel's theorem does not apply to God. Rather, I would say that Gödel's theorem is itself contingent on God. Consider how God identified Himself to Moses in Exodus 3:14: Simply "I AM."
Happy New Year to you too Mark! Great to hear from ya yesterday.
Perry
P.S.: There's a lengthy conversation with a guy named Derek on my other blog about something very closely related to this, it's at http://www.cosmicfingerprints.com/blog/incompleteness/comment-page-1/#comment-3241 and http://www.cosmicfingerprints.com/blog/incompleteness/comment-page-1/#comment-3725
I loved the article until it stop being descriptive and tried to argument for the existence of god. You made some really big jumps there, pal.
I liked your approach to explaining the existence of god, but you have to be honest with yourself.
At least you are not the kind of christian who believes Jesus used to ride a velociraptor.
cheers!
You're welcome to explain exactly where, logically speaking, I made "really big jumps."
It seems that you have grossly misunderstood Godel. There are actually two theorems.
Number 1:
"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,[1] but not provable in the theory (Kleene 1967, p. 250)."
Number 2
"For any formal effectively generated theory T including basic arithmetical truths and also certain truths about formal provability, T includes a statement of its own consistency if and only if T is inconsistent."
(Pulled from Wikipedia.)
These do not apply to all systems. They only apply to those that express arithmetic.
If logic and arithmetic describe the universe, then Gödel's theorem applies to the universe.
If logic and arithmetic do not describe the universe, then scientific thought also does not apply to the universe.
Everything I have said here is contingent on science and math being valid tools for studying the physical world. My conclusions here are therefore as valid as the practice of science itself.
I do concede that the practice of science is based on, literally, FAITH, that the universe is rational. By the way, the notion that the universe is rational originally came from Judeo-Christian theology. See http://www.cosmicfingerprints.com/blog/faq/#christian
"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."
My point was that this is not what Godel's theorems say at all. The theorems say nothing about having to assume the existence of anything outside the 'circle.' What you have done is taken a metaphor and extended it far beyond its bounds, and the metaphor was incorrect in the first place.
Derek,
Quoting you, the theorem says:
"…there is an arithmetical statement that is true, but not provable IN the theory." (Emphasis mine.)
This means that is axiomatic. It relies on an axiom, which is something you know to be true or have to assume to be true but which is unprovable. If it is consistent it is incomplete. Anything that proves its consistency is outside the statement. Therefore there is something outside the statement.
"This means that is axiomatic. It relies on an axiom, which is something you know to be true or have to assume to be true but which is unprovable. If it is consistent it is incomplete. Anything that proves its consistency is outside the statement. Therefore there is something outside the statement."
Perhaps some clarification will help. All formal systems are axiomatic. A proof is when you show some statement to be a result of the axioms. So such a statement cannot rely upon an axiom, per Godel's theorem. Also you are assuming that there is something that does, in fact, prove it's consistency, which Godel's theorems do not assert.
You also are confusing objects of language with the language itself. Just because parts of the universe can be described by mathematics, does not make the universe a formal system. It's like calling a chair a noun. 'Chair' is a noun, but a chair is not; only words can be nouns.
Derek,
If you don't think the universe can be accurately described by mathematics – and that it is not a formal system – you are free to take that position. At the same time you are then taking a position that science is a questionable endeavor.
If a statement is a result of an axiom then it relies on the axiom. And this is what Gödel is saying.
Rolling with your chair analogy, I am not saying a chair is a noun. I'm saying that a chair is an object. The word "chair" is a noun.
In mathematics, all systems rely on axioms – assumptions which must be taken to be true but cannot be proven.
If the universe is a formal system then the universe similarly relies on *something* which must be taken to be true but cannot be proven. And the thing that the universe relies on is a something, not a nothing. Therefore the universe is not the only thing that exists. There is something outside the universe which is not a system.
Wow, thought provoking post. It's not often that I see proofs of Gods existence of such high quality. That said, I'm curious to know how your take on two points that didn't quite jive for me.
First, does Godel's Theorem apply to itself? Can it truly ever be proven, if it's making assumptions that it can't prove? If it can, then the theorem is proven useless, and if it can't, then how can anything in the universe ever be proven? (If nothing can be proven, what evidence for a god could we have?)
Secondly, using the circle analogy, if there is something outside of the largest circle that it is possible to draw, then that something must be infinite. If it is infinite, then it is everywhere. If it is everywhere, how has evaded detection of scientists for so very long? If a creator is boundless, then science must have found evidence of its existence, and if not, it could be encompassed by a circle. Paradoxical…
Again, thanks for a well thought-out post.
Gödel's theorem does rely on assumptions you cannot prove, in the sense that Gödel expresses his theorem in Peano axioms, a mathematical system which is not provable within itself. Incompleteness is proven in the same way that geometry proofs are proven. It is 100% consistent with all that is known.
God IS everywhere. Judeo-Christian theologians have been saying that for 4000 years. Why does God evade detection? Because God is immaterial and we can only detect material things with the scientific process. But God is inferred in innumerable ways.
Dire misunderstanding of Godel's theorem. It was all going so well until you took the metaphor you'd constructed around the maths and tried to apply it to the universe. As another commenter has said, the theorem applies only to mathematical systems. Specifically those systems that are complex enough to derive arithmetic from.
The real "incompleteness" is that when working with the axioms of those specific formal systems, you will find there are either things that are true that you cannot prove, or things that are paradoxical that you can prove – the system is either incomplete or inconsistent.
You insist on saying that the universe must be a formal system for it to be described by science, the truth is that science describes the world _using_ various formally phrased "laws", but the universe itself is the "outside of the circle" (outside of our descriptive system) to point to in this case.
Our formal systems don't define the universe; they're a best approximation to what we observe, and the self-justifying element at the foundation of it all is the bald fact of the way things are. Reality doesn't derive from axioms, it just is what it is, and because of that it doesn't in any sense match up to what the Incompleteness Theorem was about.
Matt,
Gödel's theorem applies to everything that reason and logic apply to.
No, our formal systems don't define the universe, they describe them.
If the universe is complete it is inconsistent and thus not amenable to scientific analysis.
If the universe is consistent and amenable to logical analysis, then it is incomplete and therefore contingent on something.
I vote in favor of science. IF the universe is scientifically, mathematically and logically describable, then God exists.
You are welcome to reject the God proposition. In so doing you also reject reason and logic and science.
"Gödel's theorem applies to everything that reason and logic apply to."
No, I believe you'll find that it applies strictly to axiomatic mathematical systems; "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"
So if you start with a set of mathematical axioms, and those axioms are sufficient to express arithmetic, then it is also true that your system is either incomplete (there are truths that it can't prove) or it's inconsistent (you can prove things that are contradictory).
Now say it with me, the universe is not an axiomatic system of mathematics. There are no axioms of the universe, and hence no logical derivation of other propositions from the universe and no Incompleteness.
Empirical derivation of propositions, maybe (although that'll run the risk of simple observational error), but I can't repeat enough; the universe is not a system of formal logic. Further, it has no need to prove itself by formal logic – the universe just is. Its nature is a simple fact, that to some extent we observe.
I also take issue with your talk of "The origin of information". The genetic 'code' is not symbolic and immaterial, it's carbon-based chemistry. Not even very complicated chemistry to be honest. You claim that 'information' had to come from the 'outside', which seems to miss the fact that randomly combined nucleotides have just as much genetic information in them as the same length of useful DNA. The only difference is that genetic material that is conducive to its own replication will do just that, replicating more than other such material until it dominates.
"All codes we know the origin of are designed by conscious beings" is a barely concealed circular argument. It fails on the count that genes are not real codes (they are molecules – talk of information being coded into them is a convenient shorthand for talking about their chemical properties) and even if I let that pass, it fails on the second count that you use the assumption of all codes being consciously designed to argue that a specific code is consciously designed without any further evidence for your position.
"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 systems must rely on something outside the system. [..] Because the universe is a system, it has an outside cause."
Once again, the natural world is not an axiomatic system, and hence not something that Godel's theorem is applicable to.
Matt,
Any system of logic (boolean algebra for example) is always definable in terms of axiomatic mathematical systems.
You are free to reject the idea that axiomatic mathematical systems accurately model the universe. But in doing so you are rejecting the very premise of science, which is that the universe is weighable, countable, measurable, and that the activities of matter and energy conform to reason and logic and mathematical formulas.
The pattern in DNA is a code. All you need to do to verify that fact is read a biology book very carefully. Study the history of the genetic code. Discover for yourself why GGG=Glycine and AAA=Lycine etc etc.
The clearest explanation of this in my book collection is by Hubert Yockey. He says: “Information, transcription, translation, code, redundancy, synonymous, messenger, editing, and proofreading are all appropriate terms in biology. They take their meaning from information theory (Shannon, 1948) and are not synonyms, metaphors, or analogies.” (Hubert P. Yockey, Information Theory, Evolution, and the Origin of Life, Cambridge University Press, 2005) http://www.amazon.com/gp/product/0521802938?ie=UTF8&tag=httpwwwperryc-20&linkCode=as2&camp=1789&creative=390957&creativeASIN=0521802938
"Any system of logic (boolean algebra for example) is always definable in terms of axiomatic mathematical systems." Agreed, but I don't see the relevance.
"You are free to reject the idea that axiomatic mathematical systems accurately model the universe." I didn't, I rejected the idea that the universe itself is an axiomatic mathematical system.
To put it in your terms, the reason we can model the universe with logical systems is that we have an actual universe "outside of the circle" to "point to". (Quotations used to express the fact that I dislike the terminology; it's an oversimplification of the theorem and makes it too easy to misapply it)
The universe as we know it acts as the ultimate axiom – if the model contradicts the universe, we know the model is wrong. It may be that some physical 'laws' that we think are true turn out to be incomplete or inconsistent in their description of the universe, but that means we need an improved model, not that the universe needs some external factor to be explained.
As for genetic codes, it's as I've said; even if I accept that it is a true code, you don't have a valid argument. Saying that all known codes have a conscious origin, and that therefore a specific code has a conscious origin, is specious reasoning. To prove that all codes have conscious origin would require you to prove that genes have a conscious origin, which you have not done.
I can accept Yockey's assertion that "the origin of life is the founding axiom of biology" — biology being the study of life it doesn't make a great deal of sense to talk about pre-biotic biology.
Pre-biotic chemistry is another matter though; chemistry and physics do not depend in any sense on the existence or origins of life, and we can very usefully study the precursors of life in the form of non-living organic chemistry, and theorise on how such chemicals might become self-replicating and come to form something we recognise as living.
Matt,
The relevance of my first comment is that Gödel's theorem applies to all things that are defined as systems.
You are rejecting the idea that axiomatic mathematical systems model the universe. You are free to do that.
But if math does accurately describe the universe, then the universe is just as axiomatic as the math that describes it.
The universe cannot be the ultimate axiom if you believe in cause and effect. The universe came into existence 13.7 billion years ago. Something had to cause it.
You are of course free to reject the principle of cause and effect. That is your decision.
If we were to build a mathematical system that perfectly describes the big bang, it would rely on some initial conditions and axiomatic statements that are not provable inside the system. Therefore if we accept the principle of cause and effect, the universe is contingent on something outside itself.
My statement that all known codes have conscious origin, therefore a specific code has a conscious origin, is not specious. Perhaps this conclusion is bothersome to you, but it's straightforward inductive reasoning. You can choose not to accept that. If you reject it then you reject induction which is the basis of most scientific propositions. My statement is just as reliable as gravity and entropy. For example, the assumption that the law of gravity is the same 10 billion light years away as it is here is not proven and probably not provable either. (Formally speaking it is not provable at all.) We assume gravity is a consistent law based only on induction.
Self replication requires a code to exist first, as John Von Neumann determined in his papers on self-replicating machines in the 1960's. All codes we know the origin of are designed. Therefore we have 100% inference that the genetic code is designed and 0% inference to any other explanation.
We understand codes every bit as well as we understand gravity. Maybe better. We create codes all the time. We can't create gravity. Thus any conclusion other than "DNA is designed" is specious.
"The relevance of my first comment is that Gödel's theorem applies to all things that are defined as systems."
Mathematical systems and logical systems that are equivalent to mathematical systems, yes. I have never denied this. But the universe is neither; it's a physical system. It doesn't derive from axioms, it can't be used to derive propositions, it doesn't fit the bill.
The closest you can come to 'deriving a proposition' from the universe is to observe its behaviour and formulate a mathematical statement to describe that behaviour. Maybe the model is subject to Incompleteness but, as you've said, that just means it needs something outside of the model to serve as an unproven given. The thing outside the model is the universe's actual observed behaviour. We can take that as an absolute axiom with respect to what our models should predict.
"But if math does accurately describe the universe, then the universe is just as axiomatic as the math that describes it."
The model has, as far as we know, been accurate up until now. I don't take that as absolute/unshakeable proof that the universe will always be consistent with our predictions. If the two things diverge we'll need to come up with a new model because they remain two different things. Using an axiomatic model to describe a thing, however accurately, doesn't make that thing, itself, axiomatic.
"The universe cannot be the ultimate axiom if you believe in cause and effect. The universe came into existence 13.7 billion years ago. Something had to cause it."
I'll take this seriously when you can convince me that a) your god doesn't equally require a cause and that b) the universe (or some n-dimensional 'higher' universe) can't be self originating in the same way as you propose that god is. If you're going to restate the Cosmological argument, I feel quite happy using the age old objections to it. (If you manage the first one, you'll still need to prove that your first cause has any attribute other than uncausedness. I know you think you've done that, you even put it bold, but that's a whole other argument to have).
You don't appear to have fully comprehended my point on the subject of "all known codes have conscious origin, therefore a specific code has conscious origin". There's no denying that the former implies the latter, but your argument is backwards – to boldly assert something about "all codes" you first need to prove that thing true of each specific code without reference to anything being true of "all codes".
To put it shortly, "all known codes have conscious origin" is dependent on each specific code having conscious origin, so when you introduce "all known codes have conscious origin" to your argument about a specific code, you're implicitly assuming that which you're trying to prove. It's not "straightforward inductive reasoning" at all, just partially obscured circular reasoning.
As a sidenote, I'm curious; why is a piece about Godel's Incompleteness Theorem on a site that's otherwise about Google-based marketing?
Matt,
Postulate: IF the universe conforms to reason and logic then it is subject to Gödel's incompleteness theorem.
The universe is finite. It came into existence 13 billion years ago. How could it give birth to itself?
I think the burden of proof is on you to show that the universe CAN be self-originating.
You can resort to an infinite regress of universes, but philosophers almost universally reject infinite regression as an adequate explanation of anything.
I cannot prove to you that God doesn't require a cause. I can only point out that at some stage in the past there is a necessity of an uncaused cause. The universe is not its own uncaused cause because everything physical is subject to causality. If you have empirical evidence to the contrary you're welcome to present it.
All codes we know the origin of are either direct derivatives of DNA (RNA, bee waggles, viruses, dogs barking, pheremone trails, animal instinct) or else they are created by conscious beings making deliberate choices (zip code, morse code, bar code, TCP/IP, every single file on your computer).
My reasoning is not circular. It begins with an open question of not knowing where the genetic code came from and observing closely how all other codes came into being. The process of creating codes is well-known branch of science taught in CS departments in major universities all over the world. There is no other known process for creating codes besides sentient beings making conscious choices of symbols based on desires and priorities. DNA reflects all the same kinds of choices, from the 4-letter alphabet to the most complex genetic transpositions.
If you examine them closely you'll find that all arguments that DNA is not designed are in fact circular. I have a much more complete presentation on this at http://www.cosmicfingerprints.com/ifyoucanreadthis.htm
I really comes down to this: If you hypothesize that codes can occur naturally, then show me one empirical example that supports that hypothesis.
Why an article about Gödel on this website? Because I'm interested in a lot more things in life than just search engines.
Perry
You don't have to prove scientifically that god exists. You can just be faithful, its a more gracious way of being a believer than trying to push your opinions in a semi-literate way to everyone else…
I recommend Soren Kierkegaard.
Something out of context: " Thus, faith is united with the truth by serving as the most extreme expression of subjectivity, and by representing the only manner in which the existing individual can accept paradox."
http://www.schoolbytes.com/summary.php?disp=term&id=234
Check this site, i haven't read it all, but it could be helpful to everyone like you. In some way, a more complex way you're trying to do what every silly creationist tries to do in a more naive way, that is, trying to push a belief into science.
Sorry for the errors, english is not my first language.
I think you are putting god outside the circle boundary everything that is real;)
"My reasoning is not circular. It begins with an open question of not knowing where the genetic code came from and observing closely how all other codes came into being."
Allow me to analogise for a moment. Imagine I were to examine all of the cats I could lay my hands on, I would find that to an animal they would all have tails. I might hence conclude that all known cats have tails, and thus that I believe all cats to have tails.
Then someone presents me with a Manx cat, to which my response is that we already know that all known cats have tails, so this cat must also have a tail. I think you are committing the same flaw of reasoning; hasty generalisation.
All human designed codes have conscious design. That much is tautologically obvious. The other class of 'codes' we have are those in the natural world. We know of no designer for these things, and it is faulty reasoning to conclude that because "all known codes" (excluding the natural ones under consideration) have conscious design, that the codes of the natural world must also have conscious design.
I submit, that the codes of the natural world, be they genetic, bee waggles, or any other, are the results of what I will term unconscious design. Brought into being by blind processes with no intent, no thought, no goal and no consciousness.
I'm talking about, of course, evolutionary processes. Amino acids form spontaneously under a variety of conditions, RNA strands have been found to assemble, using cyanide or the checker-board patterns of charge in ice as a template. RNA is also known to be potentially self-catalysing, and self-replicating. Once we have replication (and mutation via faults in replication), we have evolution. Some RNAs go on to produce proteins and DNA, which turn out to be advantageous because they can improve the process of replication and help the RNA that produces them to dominate the available resources.
That path may be hopelessly wrong; we may later find the evidence that shows it happened a different way, I entirely accept that possibility. But at the same time, having any plausible explanation that fits within the natural world without any awkward 'nomological danglers' hanging out is infinitely preferable to resorting to the supernatural as an 'explanation', because it does not really explain anything at all – no predictions to be made, no evidence to be found, no proof or disproof except the incredulous cries of "well how else could it happen".
In your other article you dismiss out of hand the idea of naturally occurring codes, without any reasoning presented beyond … well, actually I don't see any reasoning except stating your position again in slightly different ways; "Chaos can produce patterns, but it has never been shown to produce codes or symbols." is another example of your circularity – assuming the general case as a way to prove the specific case, when the general case is dependent on the specific case. Maybe chaos has been shown to produce a code, in the instance of naturally occurring codes. Nothing you can say about "all codes" can rule that out.
[Sidenote: I would take exception to natural selection being characterised as chaotic; mutation is the only random step, everything beyond that is much more deterministic, except for true accidents of nature like the first carrier of a novel allele having the misfortune to be hit by a meteor or some such]
If you could prove to me, without any reference to "all other codes" or other generalisations that are dependent on the specific cases, that DNA as an information carrying molecule could not possibly arise by natural processes, then I would be most impressed. (Note also, that stating "Information itself is a separate entity on par with matter and energy" as a premise, without some very strong supporting evidence for that extraordinary claim, will not get you very far).
It's late, I'm tired, but reply to this and the email notification will remind me to come back and say more about the universe as opposed to DNA.
Matt,
Code is defined as a system of symbols for communication between an encoder and a decoder (Claude Shannon, 1948).
Show me a naturally occurring code. Of ANY kind. Just one.
The problem is that you've defined DNA as a code. You were going on about cause and effect; DNA is a series of molecules (physical things) that act in concert to produce/maintain/whatever else (effect) creatures. It's not a code; it's a chemical reaction.
Second, you are squirreled down in semantics in a big way. 13.7 billion years ago, we SUSPECT that the big bang occurred because that fits the available data. We suspect, but don't know. You say that something must have caused it. I ask why?
In university, I used the reaction/proaction argument to "prove" the existence of something beyond what we can know — i.e. if everything is a reaction, and you go back far enough, there was a beginning, therefore a proaction, therefore that energy or whatever it was…et cetera et cetera. But it doesn't prove anything. Here's why:
1) I postulate that before the big bang, there was nothingness, and in that state of nothingness some energy (God?) acted on the speck that was our universe and caused it to explode into what we now see. I postulate it, but there's nothing to indicate this is so. There's no trace of this energy now, and there's nothing to indicate it ever happened.
or
2) I postulate that the "universe" is an ineffably vast concept that can barely even be perceived by our fallible minds and senses, much less understood to any great degree.
or
3) I postulate that because thoughts are formed as words, and words can be traced back to some primitive articulation that was created by a fallible mind, all words are most likely wildly inaccurate at best, and completely fallacious. Therefore, any discussion about great concepts is like pissing into the wind. You feel good about yourself, you get it out of your system, but it ends up all over your face.
If you accept the notion that we have an imperfect understanding of the universe, then you have to accept the notion that Godel himself was imperfect. You can't state with equanimity that his imperfect theorem. based on a imperfect system, which imperfectly describes the universe can POSSIBLY be any type of legitimate proof for anything.
You can say that it is a logical confirmation of something you would like to believe anyway. But I can say (more accurately) that what we think of as logic, is most likely flawed, inaccurate and wrong. It's simply the best system we have, to date.
BB,
The definition of DNA as code is the most fundamental definition in all of biology. If you want to throw everything we know about genetics in the dumpster, that is your choice. But a more anti-scientific statement could hardly be uttered. I can only urge you to study the history of the genetic code and find out why the pattern of base pairs in DNA is, in fact, a literal code.
Your (1) is actually quite reasonable, and most importantly, pretty much matches the data we have on hand. But it still requires an antecedent event, namely an External Agent.
Your (2) avoids (1) in favor of irrationality. It abdicates to confusion. It discards reason and logic. Which in my opinion is a huge step backwards. It tosses 3000 years of philosophy and mathematics in the dumpster.
Now what I want you to notice is that your (1) and (2) summarize Gödel's theorem perfectly. Either the universe is consistent, in which case it is incomplete (God is necessary). Or else the universe is complete (no God necessary) but therefore inconsistent.
In other words if we take God as axiomatic, reason and logic and science are allowed to proceed. Then we successfully assume that we can study the universe and learn more and more. If we exclude the possibility of God then we have no choice but to assume the universe is an ineffable mystery.
Correct me if I'm wrong but you appear to be saying, "You just gave me a logical set of statements that show the necessity of God. I prefer illogic."
I would invite you to open yourself to all the different possibilities and simply follow the evidence where it leads.
Your (3) is an epistemology that we would expect Darwinism to produce. If we are the result of nothing but random copying errors filtered by natural selection, then we have no reason to suspect that our minds can accurately model the mysteries of the universe. If, however, living things have been purposefully programmed, we might rather expect that we were designed to comprehend our creator.
No, that was not was I was saying. My postulates were all statements that were equally valid (they have nothing physical to support them but they are logically sound). Any one is as likely as the other.
With regard to my (1), it does not 'pretty much match the data on hand.'
Do you know what the scientific consensus is for what happened "before the big bang?"
We don't know.
Either the universe itself has always been, or it's cyclical, or there is God, or the big bang never even happened, or or or…
There is problem in saying things with certainty when there are so many unknowns and barely-knowns. But here's one thing that is known:
Godel's Theorem isn't a scientific proof. It's a thought experiment. It's an observation about the mathematical system; but it's a THOUGHT experiment.
It wasn't that Godel went out and, say, observed anything. He simply observed that, in terms of arithmetic, there were unprovable assumptions made.
The thing is, that assumption doesn't necessarily hold true in a macrocosmic way. It's like saying, "I've observed (x) about a piece of fruit, therefore (x) must be true about the supermarket, because the supermarket contains fruit."
It's not necessarily true. People haven't disproved his theorem because it's a logically sound statement. But it doesn't necessarily hold true as you scale up (similar to how Newtonian physics is true for a vast majority of all cases, but at extremes, it becomes inaccurate).
If you want to dispute that by saying "logic is logic" or something similar, or dismiss this again by saying that I'm throwing out science, be my guest. But that's not the case. Logic is a tool and not the bottom line. And science is filled with instances of microcosmic observations being true and macrocosmic observations being false (and vice versa).
The universe cannot be cyclical because of entropy. Once a candle is burnt you can't light it again.
There is no such thing as a scientific proof. Science can only infer. It cannot prove.
Gödel's theorem is a mathematical proof. In philosophy, a mathematical proof is about as close to an absolute truth as we can get.
Getting hit by a bus and dying is an absolute truth. But it's not a scientific theorem, it's just an empirical fact.
Also, DNA is described as a code, but is certainly not DEFINED as a code. Similar to the way many people call every bit of music "songs." It's a convenient way to be understood, but it's not the most accurate.
BB,
See "Information Theory, Evolution and the Origin of Life" by Hubert Yockey (Cambridge University Press, 2005) http://www.amazon.com/gp/product/0521802938?ie=UTF8&tag=httpwwwperryc-20&linkCode=as2&camp=1789&creative=390957&creativeASIN=0521802938
Also see http://www.cosmicfingerprints.com/blog/faq
you are using a theorem within a system to explain something beyond the system. whut
using something we know to describe something which we cannot know is just faith. you jump from logic to faith when you use the incompleteness theorem to make a logical induction about something beyond "everything". you "believe" that it applies to something beyond "everything".
in other words: beyond the scope of the universe you cannot with certainty apply knowledge that is within the universe (otherwise it would be describable by the universe and therefore part of it)
Tim,
What you're really describing is the inherent problem of induction. Induction can give you only so much specificity; that's why when you apply Gödel's theorem to "whatever is outside the universe" you get a list of what God is not, rather than a list of what God is. God is not material, not matter & energy, not time, not divisible, not a system. And by the way this list is remarkably similar to Aquinas' 5 statements about God. This summary of Aquinas is from Wikipedia:
Concerning the nature of God, Aquinas felt the best approach, commonly called the via negativa, is to consider what God is not. This led him to propose five statements about the divine qualities:
1. God is simple, without composition of parts, such as body and soul, or matter and form.[60]
2. God is perfect, lacking nothing. That is, God is distinguished from other beings on account of God's complete actuality.[61]
3. God is infinite. That is, God is not finite in the ways that created beings are physically, intellectually, and emotionally limited. This infinity is to be distinguished from infinity of size and infinity of number.[62]
4. God is immutable, incapable of change on the levels of God's essence and character.[63]
5. God is one, without diversification within God's self. The unity of God is such that God's essence is the same as God's existence. In Aquinas's words, "in itself the proposition 'God exists' is necessarily true, for in it subject and predicate are the same."[64]
If you sift through every single element in both my Gödel article and Aquinas' statements with a fine-toothed comb, you can verify for yourself that it is all 100% logical.
One of the realizations that you come to is that God is "outside of" reason and logic. This at first seems disturbing until you remember that God is not divisible, God is not made up of "moving parts" and since reason and logic are about relationships between separate parts, this therefore does not make God illogical. It just makes God superior to logic by virtue of being One.
Many people have said, "Logical statements cannot be made about God." But that statement itself is a logical statement concerning God, therefore it is self-contradictory, therefore it is not true. Logical statements CAN be made about God. And the most logical statement you can make about God is:
God IS.
Thus it makes perfect sense that Jesus, who claimed to be God, said:
"Before Abraham was, I AM."
what youre saying is that the same rules of logic that apply within the universe apply outside the universe. you cant say this with certainty. the applicability of the theorem ends with the "edge of the universe" because it is based on logic of the universe.
in short:The universe and not the universe are not the same. "Universal" logic cant apply to outside the universe by definition
Which specific rules of logic do you think do not apply outside the universe?
For example:
If we draw a circle around all the matter in the cosmos and say that there is no matter outside that circle, are you saying that's not a true statement?
Dear Perry,
Thank you for the interesting read on Goedel.
May I refer a small comment to your statement:
"In 1931 this young Austrian mathematician, Kurt Godel,
published a paper that once and for all PROVED that a single
Theory Of Everything is actually impossible."
To my understanding Goedels incompleteness theorem can be
stated in simple english as:
A formal system of sufficient complexity is either
or .
I assume your above statement refers only to a contradiction-free systems,
which is indeed incomplete according to Goedel. Am I correct?
Well, my comment is the following:
If you consider for a moment a contradictive system with the potential of completeness,
would this not possibly allow the existence of a theory of everything? Of course
such a theory would then include contradicitve statements…
Here I do not want to think about such a contradictive system or discuss its
usefullness. But refering to your above statement I would like to mention,
that in my understanding of Goedel, there is the possibility of a single Theory Of Everything.
Even so it would (according to Goedel) necessarily include contradictive statements.
I might be missing a point
or miss-understanding Goedel, but at least I do see this possibility of a contradictive theory.
However I admit, that it might be never understandable by the human mind, because of its
contradictions. Would be nice, if you could let me know your opinion on this comment.
Thanks & Best regards,
Starbucks
Gödel is saying that if a system is complete, it is inherently contradictory. "This sentence is false" is a perfect example of a complete system that is contradictory.
So yes a theory of everything is possible as long as it's irrational.
This is just an interesting way of saying that if we accept that the universe is irrational, then atheism might in fact be true. Atheism can only be true if the universe is irrational.
Which is a very interesting conclusion, because the usual atheist argument is that religion is irrational and atheism is rational. Nobody prattles on and on about the superiority of their "reason and logic" more than atheists. Yes, if you are willing to be irrational then you can embrace atheism.
Dear Perry,
Thanks for your kind answer! I agree with you,
I was just curious about the possibility of an *upside down* approach on Goedel theorem using a contradictory system that gains completeness.
Best,
Starbucks
So this is still sending me reminders… which is sort of useful; I'd forgotten about it. I wanted to jump back into the fray to say that you appear, once again, to be conflating the universe with our model of the universe.
Let's say I accept your premises (you know I don't, but for the sake of argument) then the conclusion is that if we came up with a Grand Unified Theory, that described all the behaviour of the universe (was complete), then it would also be possible to derive a contradiction from that model, i.e. model a situation where our one model can be used to predict two (or more) mutually exclusive events.
That is very much like self-referential statements like "This statement is false", which tells you that the statement is both false and true at the same time depending on whether you take it at face value or think it through some more, and many other "paradoxes" where you look at it one way to get one result and then think through another line of logic to get a different, contradictory result.
The difference with a model of the universe is that we can go and determine by experiment which of the contradictory results is the right one. The model may have included a contradiction as an artifact of being made complete, but we still have an external reference to go to – empirical observation of what really happens.
I think I'm beginning to see that this is a little like the Cosmological Argument, dressed up for a new era in some slightly dodgy application of mathematics. If you'd allow me to paraphrase; "All systems must have something external to themselves, and an infinite regress of systems is impossible, therefore there must be a super-system with no external factors, and this all men call god"
It's a nice variation on the old 'first cause' or 'unmoved mover' arguments, but with the one tiny flaw that the universe is not a system of logical propositions. Godel said nothing about physical systems, and there's nothing to stop a physical system (such as a universe) being self contained.
Now I've been writing too long and made myself miss breakfast… must be off.
Matt,
As I have said to others, you are welcome to reject the idea that Gödel's theorem applies to physical things. You are free to say the universe is not a system of logical propositions if you wish.
The only problem is, you kick the legs out from under science itself when you do that. Because science assumes the universe is rational and mathematical and logical.
If E really equals MC2 and if F really equals MA then Gödel applies to the universe and the universe is incomplete.
And yes, you're right, there's that tired old old prime mover argument again. Reason and logic are troublesome things, aren't they? Whether you approach the question from a physical cause and effect perspective, or an information theory perspective, or from a philosophical perspective of regression of causes, or via moral argument, or from a mathematical perspective, you keep running into the implication of an intelligent First Cause.
If atheism is true, then the universe has to be irrational and illogical. You can choose to believe that if you want. But don't accuse Christians of being illogical. This thread has been going for months and nobody's shown any flaw in my logic.
I choose reason and logic. They lead me to God and I accept that. I likewise invite you to follow the evidence wherever it leads.
Your logic is very close to flawless. I very much liked this article, and honestly, impressed.
However, I do have an issue. I haven't checked on the formal proof of the theorem yet, but I'll take your word for it. You have an unwarranted assumption in your argument.
You assumed that whatever is outside the universe is boundless, and that is the assumption. If the universe is explained by something outside of itself, then that thing is also explained by something outside of itself, and this process would never stop, and we have an infinite amount of causes, and not one single cause. You have no clue or proof that suggests that we should assume whatever caused the universe cannot have a circle drawn around it. If the universe is any clue, circles will be drawn infinitely. Maybe the universe is part of a finite multi-verse that is part of a multi-multi-verse ad infinitum.
In summary: You proved that every ring of the chain needs a ring ahead of it, but then you assumed the chain we're connected to is infinite, which is in no way a given. Looking at the ring we are, I say it is more likely that this chain is simply infinitely long, with finite rings.
A process of cause and effect that never stops is called "infinite regression" and is almost universally rejected by philosophers. Look up "infinite regression" on Wikipedia for clarification and why that is not logical or acceptable. An ad infinitum scenario is not rational.
If universe(s) are infinite, we have no evidence of them. Everything we know about cosmology says the universe is finite. And if the universe or even a past series of universes are finite, then they are incomplete and require a transcendent cause.
Perry,
Having done a rather extensive amount of research on the subject and having discussed the matter both with mathematicians and philosophers, the concensus is that you've extrapolated to beyond what Godel was doing in the first place.
His First Incompleteness Theorem stated "Any adequate axiomatizable theory is incomplete. In particular, the sentence 'This sentence is not provable.' is true but not provable in the theory.
What he discovered and formalized was –
For any sentence s,
(1) is in PROVABLE iff s is provable.
Since the set of axioms is computably generable,
so is the set of proofs which use these axioms and
so is the set of provable theorems and hence
so is PROVABLE, the set of encodings of provable theorems.
Since computable implies definable in adequate theories, PROVABLE is definable.
Let s be the sentece "This sentence is unprovable".
By Tarski (Undefinability of Truth Theroem), s exists since it is the solution of:
(2) s iff is not in PROVABLE
Thus
(3) s iff is not in PROVABLE iff s is not provable.
Now s is either true or false.
If s is false, then by (3), s is provable
This is impossible since provable sentences are true.
Thus s is true.
Thus by (3), s is not provable.
Hence s is true but unprovable.
Godel discovered that the sentence "This sentence is unprovable" was provably equivalent to the sentence:
CON: "There is no with both and in PROVABLE".
CON is the formal statement that the system is consistent.
Since s was not provable, and since s and CON are equivalent, CON is not provable.
Thus –
Godel's SECOND Incompleteness Theorem:
In any consistent axiomatizable theory (axiomatizable means the axioms can be computably generated) which can encode sequences of numbers (and thus the syntactic notions of "formula", "sentence", "proof") the consistency of the system is not provable in the system.
The theories of real numbers, of complex numbers, and of Euclidean geometry do have complete axiomatizations. Hence THESE THEORIES HAVE NO TRUE BUT UNPROVABLE SENTENCES. The reason they escape the conclusion of the first incompleteness theorm is their inadequacy, they can't encode and computably deal with finite sequences.
So… it's a mischaracterisation AT BEST to try to use Godel's theorem to extropolate the existence of god because, if math and science is correct, which we have every reason to believe, our universe HAS NO TRUE BUT UNPROVABLE FACTS. The notion of science is that at some point, everything is explicable.
And scene.
The problem with your argument is your final statement:
"The notion of science is that at some point, everything is explicable."
Gödel shows that if it is explicable (consistent) then it is necessarily incomplete.
Science itself is incomplete. It cannot explain itself. Historically, science came from the theological proposition that the universe operated according to fixed, discoverable laws. Believe in God was necessary for science to even get off the ground. More at:
http://www.cosmicfingerprints.com/blog/faq/#christian
Perry,
You could take away the final statement and you're still left with the one before it as a problem with your argument:
"…if math and science is correct, which we have every reason to believe, our universe HAS NO TRUE BUT UNPROVABLE FACTS."
Science is based on the assumption that the universe is complete in and of itself. Similar to the way Euclidean Geometry, real numbers and complex numbers are not subject to Godel's Theorem; if the universe is a complete system, it too is beyond the scope of Godel's theorem.
Godel's theorem deals with the paradox inherent in an incomplete system. It doesn't deal with labelling complete systems incomplete and then ipso facto, there's something outside the system.
Ben,
Science does not assume that the universe is complete in and of itself. In fact science makes all kinds of unprovable assumptions, such as the assumption that the universe is logical.
You cannot truly prove that, you can only demonstrate that it's a workable hypothesis.
And I'm sorry but you're mistaken, Euclidean Geometry, real and complex numbers ARE subject to Gödel's theorem. That's the whole point of the theorem. That every system of mathematics is incomplete.
Syllogism:
1. Gödel's incompleteness theorem applies to all logical systems.
2.The universe is logical
3. Therefore the universe is incomplete.
Godel himself postulated that his theorem did not apply to Real and Complex number systems as they contained no paradoxical statements.
Are you suggesting that you have a better understanding of his theorem than he himself did?
In addition, science DOES assume that the universe is a complete system. It also assumes that the universe is describable. Science is based on all sorts of assumptions. Fortunately, SCIENCE ITSELF is an INCOMPLETE SYSTEM that depends on THE UNIVERSE for completeness.
I need you to quote Gödel's exact statement regarding real and complex number systems.
If the universe is logical, then the universe also is incomplete.
I think a significant problem with this discussion so far has been an ambiguity with the language used, which has meant a bit of back and forth with no real change in the arguments. There have also been some misunderstandings of Gödel's theorem, which is understandable, as unless you have studied mathematics, analogies tend to obscure the strict meaning of it.
Firstly we can agree that a non self-referential statement is either exactly true, or not exactly true. Secondly Gödel's theorem refers to a set of statements, split into two groups, those we label "axioms", and statements derived from those. The theorem essentially states that if the "axioms", which are the statements we assume to be true, are true, then there are also other true statements that cannot be derived from those axioms, and that those axioms cannot be proven to be true without the addition of a new axiom. The corollary is that if a proof of the truth of the axioms is derived from the axioms, then the axioms are not true (in their entirety). I'm going to use a bit of an analogy for this next part, if you can forgive me for that, as I personally find analogies highly dubious. In mathematics sequences of numbers can be of particular interest, and these can be generated through formulae, or can exist independent of any formulae. Now what is particularly interesting is that for any finite sequence there are an infinite number of different formulae that can generate the same sequence, so when attempting to find the formula for generating a sequence you may have found one that matches all available data, yet not be the correct one. I guess that wasn't particularly an analogy so much as a cut-down version of universal laws, however that is not the point I am aiming to make. Within this analogy the universe is the sequence of numbers, the formulae which determines it's behaviour are the laws of the universe, and our estimation of the formulae are our current laws of physics. Now the point that Matt was making, and perhaps you misunderstood, is that the sequence of numbers, and the formulae are different things, he was not arguing that Gödel's theorem did not apply, merely reminding you that it only applies to sets of statements i.e. the formulae, not the sequence, the laws of the universe, not the universe itself. He is not arguing against logic, merely stating that you have applied it in the incorrect place. Now as it happens I am an atheist, I believe that there is no God, however I am not so presumptuous as to claim it as a certainty, indeed I do not believe it is possible to know whether or not there is a God. I am merely arguing that this particular path of logic is invalid.
With regards to causality, it can be broken down fairly simply, either everything requires a cause, in which case a being without cause does not exist by definition(granted this does not forbid a being with a cause which created the universe from existing), or not everything requires a cause, in which case the universe does not demand the existence of a God without cause(although one is possible). With regards as to codes, we really need to clarify what you exactly mean here in your argument, as if you define a code to be a message intended to send information, then all codes are the work of sentient minds merely by definition.
Gödel's theorem is about systems of logic.
I cannot prove that the universe is a logical system. However, the assumption that it is is the cornerstone of modern science. If the universe is logical then it is also incomplete.
I define communication systems per Claude Shannon's 1948 paper "A mathematical theory of communication" which applies to things like telegraphs, CD players, computers, cell phones, etc.
All communication systems we know the origin of are designed. No known exceptions.
Sorry but 'god' must also have something 'outside its circle'
Godel proved that you cannot have a god which explains himself.
Atheism also says that.
NOTHING can understand it all. God CANNOT exist.
Bob,
If NOTHING can understand it all, how is it that you can then say "God CANNOT" exist?
As for your first statement, you don't appear to have followed the logic of my argument. You need to substantiate your statements with logic.
Yea, so I have no problem with your logic in saying that "god" exists, but I do have a problem with the assumption that this god is the christian god. Every logical proof of "god" simply defines god as something undefinable. That does not, in any way, mean that the christian god is the god that created the universe. Religion is made up to answer unanswerable questions (or at least unanswerable at the time of its inception) and as a means of social control, and to give hope, and meaning and blah blah blah.
Also, almost every response you've made to people goes along the lines of, "If you think that, you obviously reject science and are an idiot." Not very classy….
But, yeah, christians are illogical for saying the god of the bible is the god that started the universe. The only conclusion you can come to is that something must have started the universe, assuming Goedels theorem applies (which, as has been stated, really doesn't apply to the universe, go ahead and tel me I'm an idiot for saying that, it's fine).
Also, I would like to question, what is the significance of all of this? Does if affect anybody in anyway? I really don't think it does. I'm just bored and want to get into this heated discussion, heh.
I'm sorry for my choppy/poorly thought out comment. ADD is not my friend this night.
Look out your front door. Turn on your TV. Look at this discussion thread.
I invite you to answer your own question:
Does the truth about God – do claims about God – affect anybody?
Still getting those email reminders every so often… odd that they don't seem to come through for all the replies, but then I suddenly get 10 at once. Oh well. Wanted to reply to this specifically: "Does the truth about God – do claims about God – affect anybody?"
If god existed and had a measurable effect on the physical universe, it would bring it under the purview of science – god would be like the force of gravity; observable, measurable, predictable. Maybe not so predictable as gravity, but we could reason about the effects of god on the universe.
Funny thing is, we haven't observed any supernatural effects on the universe; nothing that can't be reduced down to naturalistic physical principles. So in a very real sense, the existence or not of god has no effect. If god does exist, he's not doing anything with any noticeable effect, so why care?
We tend not to believe in the existence of things with no observable effects though; that's the only way to keep obvious absurdities like invisible goblins and Russel's teapot out of our belief system.
Another sidenote; your website remembers my name and email perfectly, but seems to have complete amnesia on the question of whether I want your "free mini-course" delivered to my email. It's the strangest thing, almost as if you're hoping repeat-commenters will eventually forget to untick the box and inadvertently sign up for your mailing. But of course you wouldn't do a sleazy thing like that, so I'm sure it's just a bug in the server.
Matt,
We have observed supernatural effects in the universe. See http://www.cosmicfingerprints.com/blog/faq for a discussion on the lower half of the page. In any large public library you will find an entire section on documented cases of miracles. Just because they're politically incorrect doesn't mean they don't exist. I describe my own personal experiences at http://www.coffeehousetheology.com/miracles
Reason concerning the effects of God on the universe is a field known in academia as theology. I invite you to set aside the pejorative attitude and follow the evidence where it leads.
Interesting article, but Matt did a very effective job of demonstrating that the universe is a physical, actual system, not reliant upon abstract axioms to prove its existence.
Sorry, but Matt is very clearly better informed on this issue. The language of the article oversimplifies the theorem so as to allow the author's worldview to seem better justified. The comments made a nice read, though.
Ben, I have two questions for you:
1. Is the universe logical?
2. Can you name any law of physics that is not expressed as mathematical or logical statement?
I haven't read all the replies above so I am probably repeating some people but below are the flaws or premises that I disagree with in your argument.
1) You define God as what is outside the largest circle. As the number of circles is infinite it is meaningless to talk about the largest circle. Gödel's theorem suggests that all things depend on something else, if a god exists and is subject to our laws of mathematics then it is no exception, therefore by use of modus tollens if a god is subject to our mathematics then it must be dependent on something else to exist.
The existence of a god is not necessary to explain how something can be true despite depending on an infinite chain of assumptions. It is clearly wrong to say that because each natural number requires you to assume that there exists a larger natural number that there must be a god that is larger than the largest natural number. The natural numbers and, if Gödel's theorem is true, the ever increasing circles that depend upon things they do not contain are infinite
2) All codes must come from a conscious being. You are using a double meaning of the word code, its normal meaning is a message from one conscious being to another that has be altered to make it difficult for those who are not its intended reader to understand. DNA is referred to as a code because it contains information that is difficult to understand, the same could be said of many things that do not originate from a conscious being, pulsars (rotating neutron stars) are a simple example, we receive from them a brief burst of radiation which we can decode to gain information about the star.
3) A being can be conscious, boundless and without cause. To me conscious seems to cause problems when combined with the following properties; if something is conscious then it is aware of itself, if it is boundless then there is an infinite amount of itself that it has to be aware of, information from its farthest edges would have to travel with infinite speed to cover an infinite distance in finite time. If it is without cause then it must have existed forever, over an infinite quantity of time a conscious being would go through an infinite amount of thoughts and would end up unable to think anything new and robotically repeat through previous states, this is not a contradiction but is probably not how most would think of a god. The same problem occurs with an omniscient god as if truly omniscient it would know in advance what it was going to think and do and at the correct time would think and perform those actions like a robot.
4) In what little of the comments you mention the Holy Trinity so I shall assume you are referring to the Christian God. This is where your argument really falls to pieces, you go from trying to prove the existence of a potential creator to the existence of three very specific Gods. There is nothing in your debate on Gödel that applies to the Christian Gods and could not apply to Ra the ancient Egyptian Sun God or to the thousands of gods that humanity has worshiped throughout history, or to any of the millions that I could invent, name and create a back-story for right now. Assuming that all other points I have made are false and your argument is sound you have still proved nothing other than a vague conscious being outside our universe that somehow proves some things in our universe to be true.
Finally I would like to thank you for the excellent quality of the first part of you argument, it is one of the best explanations of Gödel's theorem I have ever read, I apologize for the length of this piece but may add more if I think of anything else.
1. There is not an infinite number of circles, because the universe is finite.
2. I am using the exact same definition of code for biology as I am for computers etc etc. This is explained in extensive detail at http://www.cosmicfingerprints.com/blog/information-theory-made-simple/ and elsewhere on my site. Pulsars are not digital commuication systems.
3. You are defining an infinite being in terms of space but I clearly explained that God is outside of space and time and I explained the logic behind this with care in the article. Got is not matter, not energy, not space, not time. Information does not travel from one end of God to another in packets. Please go back and read my statements with care. Your statements about infinite amounts of thoughts are self contradictory since any infinite sequence by definition never repeats itself.
4. The Trinity does not say there are three specific Gods. Please do not misquote me, and if you wish to critique the theology of the trinity then please avail yourself of theologians who have written about it such as Augustine, Aquinas, Jonathan Edwards etc. Or read exactly what I have written: That God is ONE and is in complete unity has three aspects: Self intent, expression of intent, and understanding of self.
Bizzare:
1- Assume the whole universe is just:
The concept of the number 1.
The concept of the number 2.
The concept of sum.
We can draw a circle around the whole universe and safely say that 1+1=2 without the need for external reference.
2-I think not. You're assuming that, using the same logic the article points as flawed (inductive reasoning). It reads "you cannot PROVE gravity will always be consistent at all times. You can only observe that it's consistently true every time", and while this logic can be used to reach some truth it can also lead to fallacies. It isn't as safe to assume that all codes come from a conscious being as saying that gravity makes things fall.
3- You're just trying to say god exists
4- You're trying to hide the fact from #3.
1. If you say 1+1=2 you reference numbers and symbols which are outside the universe you just defined. In your own description you are standing outside of the universe. The whole thing still depends on you.
2. Show me a naturally occurring code. Just one.
The "unknown" agent outside the circle always somehow seems to point to one's favorite religion or metaphysics (the immaterial vs. material). The true incompleteness perhaps lies in the symbolic, linear conceptions human understanding is stuck with. This theorem FALSIFIES it does not give positive verification to speculations or personal convictions. It is much too easy and sloppy to interpret the limits of understanding as reflected by our greatest proofs as an excuse for thought termination and to fill in the cosmology of the unknown and unreachable with all too human answers.
As for "code" the human mind has a tendency to see patterns everywhere, even in vacuum energy. Using this as proof of supernatural intelligence (since these things are everywhere in the universe) violates the very theorem you are misusing (the circle would prove itself). Intelligence is itself recognized through quite finite and definable parameters that are always experienced well within the circle. If human intelligence and the tragic world of crude matter and its pointless replication and destruction are really some positive indication of what lies beyond, a perfect God would have to be exchanged for a perfect monster, which would have to "inductively determine" from the very cruelty of nature. A cruelty, by the way, which intelligence only seems to make MORE insanely devious, not less. God should do us a favor therefore, to remain an Impersonality.
Codes are not a subjective impression on the human mind, they objectively exist. DNA is decoded by the ribosomes whether we see it and acknowledge it and describe it or not.
I suppose various people have tried to tell you that God is some kind of cosmic teddy bear, but you surely did not get that from me. Indeed, surely if nature tells you anything, it tells you that God can be wild and ferocious. If nature tells you anything about God, it also tells you that God can be soft, tender and beautiful.
Open your eyes, look around you, and see.
And as you do so, do not neglect to distinguish the difference between what God has created and what man has destroyed.
I would invite you to open yourself to understanding God as God really is, and follow the evidence where it leads.
"Codes are not a subjective impression on the human mind, they objectively exist. DNA is decoded by the ribosomes whether we see it and acknowledge it and describe it or not."
Their physical reality is precisely why describing the process as a code is flawed. Take a symbolic system like writing, we take information from our thoughts and ideas and encode it into letters, words, sentences, all according to the rules of our language. None of these odd little squiggles on paper or a screen make any objective sense, the information can only be retrieved by conscious understanding of the symbols and the language.
DNA is different; the chemical properties of the molecule are what allow a ribosome to take a string of DNA as a template for building a protein. The base pairs quite naturally have some affinity for each other, and amongst the millions of chance collisions between molecules in cell chemistry that lets the ribosome find the right strand of tRNA to pull an amino acid from to build a protein.
I'm not a molecular chemist, so I don't know every detail… but if I remember rightly I think the 'affinity' stems from the shape of the amino acid being conducive to the formation of hydrogen bonding. It's not too horribly complicated but it's been a long while since I've had cause to think about it.
The important part is that it all happens chemically; the ribosome isn't required to understand what it's doing, it isn't "decoding" the DNA in any real sense – it's just a protein that reacts with DNA and RNA molecules in such a way that some other protein is formed.
The defining difference is that our symbolic codes have no objective relation to the information they represent. Nothing really links the letter A to an "ah" sound, or any of our words to their meanings, except for our minds and the associations they store as language. With the genetic 'code' the DNA molecule and the protein synthesised are related by deterministic chemistry. You could decide to redefine a word, you couldn't redefine a DNA sequence.
GGG is not Glycine, it is instructions for making Glycine. AAA is not Lycine, it is instructions for making Lycine. DNA has start bits, stop bits, error correction mechanisms, pointers, objects. CD players don't "understand" what they're decoding either but they decode just the same.
If you want to debate this, you may do so on the Cosmic Fingerprints website. But you will need to do your research first – this ground has been thoroughly covered on my website for 4+ years now. See http://cosmicfingerprints.com/dnanotcode.htm, http://www.cosmicfingerprints.com/blog/faq. These definitions have been universal for 50 years.
In short, you will get nowhere with this argument. DNA is just as much a coding system as your hard drive. In fact the list of similarities is long indeed.
Perry,
We think your main misinterpretation is that you believe Gödel's theorem applies to the physical universe. You say, basically, that Godel's theorem applies to science because science is made of logic. Therefore, science is either incomplete or inconsistent. We agree. But because humanity can change what it believes, science adapts to evidence to remain consistent. This leaves only the possibility that science is incomplete. We think you will agree with this.
But you then say that Godel's theorem applies to the Universe and/or Multiverse, because it can be described by science. Wait, didn't We just say that science is incomplete? That would mean that science does not fully describe the Universe/Multiverse, and therefore is in some way wrong. So Godel's theorem cannot apply to the universe, because we do not actually know it is a system. Even if we did, we would not know it's limits.
Additionally, Godel's theory could be applied to itself if we use your 'put something in a circle' interpretation, which is wrong anyway. If we did this, it would prove itself wrong, meaning you entire argument is, in fact, invalid.
You need to pick an argument and stick with it.
Gödel's theorem does not prove itself wrong because his theorem is also based on axioms. For example the assumption that peano arithmetic is valid. He wrote his original proof in peano arithmetic.
You're right, science is also incomplete. No matter how much we know about science, if it is consistent it will require something outside of itself to verify its truth. There we go again – eventually we need Something that is outside the universe.
Science ASSUMES that the universe is logical. It cannot prove this. If the universe is logical, it is incomplete.
If you draw the biggest circle around everything, you're not leaving anything outside, you have from the explainable to the unexplainable inside, it doesn't matter if it can be explained or not, everything is inside thus there will be no need for outside reference.
What this theorem seems to do is first to expose some really good logic, it certainly is right in smaller scales and/or when using the adequate example (the bike for instance), once it has proved that this reasoning can work it proceeds to use the same statement with everything else, assuming that if it worked there it must always work.
I have to admit I was agreeing with it, but the part about inductive reasoning clearly reveals a flaw. While it is true that inductive reasoning can lead to mistakes, it isn't the case every time, because you can prove things, even if you start looking at the facts from the inner circle. Gravity being of those provable ideas. You also have to keep in mind that some things, when explained or researched upon, give more questions than answers, and this is another issue used to support this Incompleteness theorem, but it's just a part of the search for an answer, and raising questions helps to nothing but to better understand everything, the more questions the closer to the truth we get.
I believe this theorem can be both flawed and perfect depending on the circumstances. And that it suffers from the same misuse of general relativity Once I had a very fiery discussion with a friend who said that everything, from matter and energy to concepts are relative, to which I replied that concepts can be constant, as units like 1 meter, the concept of 1 meter can not be relative, maybe the tool used to measure differs, maybe it has shorter centimeters, but the concept will not change under any circumstances. Perhaps this is the case.
I want to finish saying that I don't reject the idea, humankind will probably never discover the secrets of everything (I'd bet to 42 tho), but I had to speak up because I think that some of the logic in the article is somewhat off.
"* Whatever is outside the biggest circle is an uncaused cause, because you can always draw a circle around an effect."
while this may be true, you can also always draw a circle around the CAUSE…therefore the cause cannot be outside the circle
"* All codes we know the origin of are designed by conscious beings.
* Therefore whatever is outside the largest circle is a conscious being."
Oh, where to start? LOL
1) Information is a thing…maybe not in the physical sense, but it still exists, and is therefore IN the circle.
2)Just because all the codes THAT WE KNOW are the creations of conscious beings in no way shape or form implies that all codes must be such creation. It only suggests that there are codes that we are unfamiliar with.
3)once again, if someTHING exists outside of the circle to generate information, or anything else for that matter, logic would dictate that you could then draw a larger circle that would encompass that creator as well
If you want to make a logical argument, it usually helps to actually USE logic and not just tiptoe around it
Graeme,
You will need to read the other responses to other posts in this thread. Every issue you have brought up here has already been addressed.
In a loose sense of the world "address". You've diligently responded to all my comments, but I'm not convinced you've really given a satisfactory answer. I and others have asked you to demonstrate that DNA really is a consciously designed code, without appeal to "all codes are consciously designed", and "DNA certainly is a code" (I could accept either one of those, but not both at once, depending on whether you take 'code' to imply conscious design by definition or not)
Fundamentally, you can't assert something to be true of all codes, then use it as proof about a specific case; to make the general statement requires the proof of the specific case. And yet, your own "proof" of DNA being designed by a mind follows as: (1) DNA is a code (2) All codes we know the origin of are created by a conscious mind. (3) Therefore DNA was designed by a mind.
Point 3 simply does not follow from points 1 and 2. You're effectively trying to establish the origin of DNA _from_ now knowing the origin of DNA. If you have a way to demonstrate the conscious origins of what I hold to be a naturally occurring code (or a non-code if you really want to assume code to imply conscious origin) then I'd love to see it, but I'm still waiting.
(And before you ask) my proposed origin for the system would be a very rudimentary genetic molecule, likely to be RNA, synthesised by prebiotic chemistry and having just barely enough complexity to get a copy of itself made. Once that foundation is in place, mutation and selection over vast spans of time could arrive at the much more efficient system we see now.
Oh, and I may have made the mistake I've been hoping I wouldn't and just clicked submit with the little box ticked. (If this is a duplicate comment via double-submission, please feel free to delete the first) No matter, a little gmail-fu will sort that one out.
New codes are created all the time. Recent examples from the last 20 years include PDF, HTML, DLL, JPG, MOV, DOC, XLS. Each of these is distinctly different from the others, using in some cases radically different coding structures. All of the above are stored / transported on dozens of different kinds of media including CD-R, hard drives, laser light, voltage pulses etc.
If the question is stated as: "Is there a natural chemical process that produces codes? Or do we have reason to believe there is some sort of Intelligence behind biology" then Point 3 naturally follows from points 1 & 2. In science this is called inference.
Please support your proposed origin theory with an experiment. Preferably an experiment that is not designed. The ideal would be an example of biogenesis from some "warm pond" somewhere.
Or show me ANY code created from scratch by any process other than design. All you need is one.
1) All known plates are designed by conscious minds.
2) The earth has tectonic plates.
3) The earth was designed by a conscious mind.
This is what happens when you play fast and loose with semantics.Just because you've used the word "code" instead of plate people get all confused and mystified, but it amounts to the same thing.
Now let's see what happens when you abuse metaphors:
1) All known sandwiches are created with the intention of being eaten.
2) The solar system is sandwiched between other galactic formations.
3) Therefore the solar system (and the starry loaves in which it is nestled) is going to be eaten.
Now let's see what happens when you can't tell the difference between a system (science, math, language, etc…) describing a thing and the thing itself (the universe).
1) "Apple" is a noun.
2) Nouns are found in the dictionary.
3) Next time you're feeling hungry, open the dictionary up to Apple and chow down (with apologies to Gwyneth Paltrow).
Now after seeing Matt's point-by-point rebuttals of your arguments and your refusal (or inability) to understand the strength of his objections, I can't assume this will have any effect on you (why, for instance, you are troubled by infinite regress but not by the idea of an immaterial god who sits outside of existence but still affects it and – what luck! – requires no cause, is a mystery on par with that of the big bang itself) But hopefully this is a simple explanation of just a few of the flaws in your reasoning for people who read your article and aren't wearing god-blinders.
The word "code" is used in precisely the same way and with precisely the same meaning in both computer systems and biology.
“Information, transcription, translation, code, redundancy, synonymous, messenger, editing, and proofreading are all appropriate terms in biology. They take their meaning from information theory (Shannon, 1948) and are not synonyms, metaphors, or analogies.” (Hubert P. Yockey, Information Theory, Evolution, and the Origin of Life, Cambridge University Press, 2005).
See http://www.cosmicfingerprints.com/blog/solve.
All philosophers are troubled by infinite regress.
An uncaused casue is not infinite regress. Which is exactly my point. At some point, one is required. Godel's theorem shows why the universe cannot be its own uncaused cause.
If you want to seriously take part in this discussion, you will need to apprise yourself of the definitions used and the details of the argument.
It's clear that it isn't used the same way. The mere fact that human designed codes are self-evidently designed by humans and the genetic code isn't is enough to show that. That there are resemblances does not make them the same. The fact that you are asserting that they are used in exactly the same way only serves to highlight your circular reasoning.
"I can take quotes out of context and misapply them too." – Me in a different conversation that had nothing to do with this.
The fact that you would try to take Yockey and use his quotes to argue for intelligent design would be comical if it weren't insulting. I won't go into depth as to why it's wrong here, because, thanks to the power and beauty of the internet, someone else has already done this for me. Here's a teaser:
"I'm told creationists have started citing this new book of his in defense of their own argument that God must have started life on earth. I've not seen this abuse myself, but it wouldn't surprise me, and at any rate, forewarned is forearmed. If that's their plan, Yockey throws a bucket of cold water on it." — To find out why you'll just have to read the article. Here's the link – http://richardcarrier.blogspot.com/2006/11/yockey-on-biogenesis.html
As for infinite regress and the uncaused cause, this is literally the kind of thing they do in intro to philosophy courses. I'm not going to reiterate the entire history of the cosmological argument and its objections, but you're welcome to read a brief introduction to it at http://en.wikipedia.org/wiki/Cosmological_argument#Objections_and_counterarguments
I guess you think that by throwing Goedel in you can make an old dress new, but just like creationists attempt to rebrand themselves as "intelligent design," it's lipstick on a pig.
Cute bit at the end there. If you want to seriously insult my intelligence while seeming dignified you'll have to apprise yourself of more intelligently designed slams.
Mike,
If you want to refute my argument about Godel, you will need to refute it line by line. Snide remarks about "lipstick on a pig" are no substitute for facts and logic.
You will use 100% respectful language in your posts from this point forward or your posts will be deleted.
Yockey on biogenesis: p. 176: "I have no doubt that if the historic process leading to the origin of life were knowable it would be a process of physics and chemistry. Thus the process of the origin of life is possible but unknowable."
"The reason that there are principles of biology that cannot be derived from the laws of physics and chemistry lies simply in the fact that the genetic information content of the genome for constructing even the simplest organisms is much larger than the information content of these laws" Yockey wrote this in his previous book (1992).
"The existence of a genome and the genetic code divides living organisms from nonliving matter. There is nothing in the physico -chemical world that remotely resembles reactions being determined by a sequence and codes between sequences."
Yockey states that the genetic code must be taken as axiomatic, much as we take Planck and Newton's axioms, because it cannot be derived from the physics of chemistry.
Pretty sure I've already addressed your use of Yockey as a pro-creationist source. I also think I've made it clear that I'm not trying to convince you personally of anything. The posts above already refuted your argument days before I even read it, the fact that you don't see that speaks volumes. You have been refuted and you don't see it, so you'll forgive me if I choose not to waste my time any more than I already have.
My commentary was hopefully just illustrative for anyone who didn't see the plain flaws in logic. I don't think I've said anything here or elsewhere to impugn you, as opposed to your arguments posted (and, as I said, refusal/inability to take any rebuttals seriously).. In terms of your arguments, arguments don't deserve to be treated respectfully, they deserve to be attacked. Those that are strong enough will withstand the attack. Yours do not.
You will use 100% non-condescending ultimatums from now on or I won't post anything else — I probably won't anyway, I just wanted to let you know in my own special way that I found that to be gratingly paternalistic. Since this is only 93% respectful, you are welcome to delete it.
I quote Yockey precisely because he's not a creationist. Same with Claude Shannon and others. 99% of the people and papers I quote are non-controversial mainstream peer reviewed research. There is nothing remotely controversial about the fact that the genetic code is in fact a code.
I fully understand that Yockey disagrees with me philosophically. There's no rule in science, philosophy or life that the person who produces the evidence you use in an argument has to agree with the conclusions you reach about that evidence before you can use it.
Yockey makes the clearest case I've ever seen that the laws of physics and chemistry do not explain the origin or nature of the genetic code. Yockey stops right there and goes no further. I continue forward with the simple observation that all codes and coding systems we know the origin of are designed. Therefore we have 100% inference to design in biology.
This is not proof, but it is scientific inference. As such it is just as reliable an observation as the laws of thermodynamics and gravity.
The fact that Yockey doesn't agree with this is no concern of mine. Yockey tells you there's no other instance of a naturally occurring code, just the same as I do.
I have taken every legitimate rebuttal seriously, as I have debated this particular topic online for 4 years now. See http://www.cosmicfingerprints.com/infidels for an entire history of this debate. You can attempt to argue that DNA is not a code in the same sense that ASCII is a code but that line of argumentation will get you nowhere. The literature is very clear on what the definition of a code is and both examples fit that definition 100%.
Mike, if you can produce one example of a naturally occurring code I'll write you a check for $10,000 and post your evidence on my website. The specification for doing so is http://www.cosmicfingerprints.com/blog/solve/
Until someone can successfully demonstrate a naturally occurring code, the only scientific inference we can make about the origin of the genetic code is that it's designed. And note that in saying so I have not stepped outside the bounds of science at all. We don't have to know who the designer is or the exact details of the design process in order for this inference to hold.
1. All types of rapid transit utilities (bikes, cars etc…) are designed by conscious minds and not naturally occurring.
2. Horses are a type of rapid transit utility.
3. Therefore, horses are designed by a conscious mind.
I will give 10 million rubles to the first person to show me a naturally occurring rapid transit utility — keeping in mind that horses don't count, donkeys, ponies mules, bulls, buffalo neither (Half credit for a midget riding a lasso apso).
Seriously man, good luck with life and everything, but I am beyond done here.
In relation to this comment…I believe Perry's point was that when you're discussing codes, there is no example of a horse.
Hy folks. Nice debate. Roy, you stated "What caused God? It's not a coherent question. By definition God is eternal…uncaused." Ok, but by who's definition? What if the nether that harbors the multiverse is what is really eternal and infinite. What if it has always been. No one can prove otherwise and and the proverbial circle cannot be drawn around it. If this were the case then God simply fades into one possible explanation as opposed to "the" explanation. What of time? We experience time on a liner scale but that is a function of the brain (as opposed to Brane…couldnt help it), and no necessarily how time exist in its own dimension. Can you draw a circle around time? Can you prove you can? No, you cant. Time may well extend infinitely in all directions unaffected by gravity, energy, velocity, expansion, or the boundaries of the multiverse. Does this prove that all things are begotten from time?…maybe. Or is time the true God? Either that or time doesnt exist and is an illusion according to the reasoning of this article. Thats been proposed before but not readily accepted by science, much less proven. I think what we have here is a creationist trying to prove his view through a very guided discussion using a single mathematical concept; Gödel's Incompleteness Theorem. To truely understand the implications of that theorem we would all need to be doctorate level mathematicians. A single simple tenet of this theorem was put in even simpler layman's terms. Then that was used to try to argue a point as absolute. Gödel's Incompleteness Theorem does not "prove" the existence of a god (much less a Christian, Buddist, Muslim, Jewish, Egyptian, Greek, African, or any other god invented by mankind). Yes! I whole heartily agree that the possibility of a cosmic conscience exists and that it might be Godly. I believe this because there is no way to completely disprove it either. If time is finite, which we cannot prove, then there is always another circle. Keep in mind. Time was just one of many examples. Replace time with an infinite multiverse, or nether that cant be circled or just consider that the larger circles are also infinite once you break past the multiverse. We have NO way of knowing and many of the variables to this conundrum and God is just one of many…it just happens to be the one you favor.
- Eric M.
My assertions are based on the facts about the universe that are known at this time.
The facts as we know them are: The Big Bang occurred 13.7 billion years ago, that it expanded from a single point, and that it has finite mass and energy.
Time and space are inseparably linked. No space = no time. Time itself begins with the big bang. Therefore time is finite. Yes, you can draw a circle around it. It has boundaries.
There may be other dimensions of time and other universes but it is impossible for us to know anything about them.
You are welcome to assert that time and even our own selves are illusory but I don't know how you're going to build any kind of epistemology from that foundation. You're on your own if you want to embrace that view.
If you want to posit an infinite number of universes, I'm going to ask you to provide evidence that they exist because invoking an infinite number of anything violates parsimony.
God is defined as eternal and uncaused in Judeo Christian theology.
Gödel's theorem directly implies that whatever caused the universe is timeless, boundless and uncaused and not a system.
Which does interestingly match Aquinas' Via Negativa.
No, Gödel's theorem does not prove God. It only directly infers the existence of an axiom which the entire universe must rest – something which is not space, not time, not matter, not energy, not a system, conscious and boundless.
"Pulsars are not digital commuication systems."
Neither is DNA. You just proved to yourself that DNA is not a code by your definition…
Brandon I suggest you consult a dictionary, in which one of the very definitions of "Code" is THE GENETIC CODE. Hubert Yockey's book should lay to rest any notion that DNA is not a communication system and I invite you to read it before continuing to make this assertion.