Comments on: Gödel’s Incompleteness Theorem: The Universe, Mathematics and God

80 years ago, Kurt Gödel toppled empires of mathematical philosophy with his famous Incompleteness Theorems.

Kurt Gödel proved, ironically, that it's impossible to prove everything. And yes, he proved it.

I’m every bit as interested in science, philosophy and engineering as I am in business. Gödel’s theorem has profound implications for every branch of knowledge.

The materialist view prevails in secular circles. Materialism states that the laws of physics and the universe we know are all that is. It sees the universe as a giant machine. It assumes that everything we experience is purely the result of blind cause and effect. It scoffs at the idea that there is any such thing as God or metaphysics.

This view was epitomized by “Logical Positivism” which was espoused by a group known as “The Vienna Circle” in Austria, led by Ludwig Wittgenstein. Logical Positivism says that anything that cannot be experimentally verified or mathematically proven is invalid.

The Logical Positivists were confident that very soon, all the loose ends of mathematics would be nailed down by a single unifying theory. The world would finally fully embrace reason and logic and leave the failures of religion behind.

Kurt Gödel was a member of the Vienna Circle and in 1931 proved that a single unifying theory was impossible. He proved that the goal of the Logical Positivists was unachievable. This was a devastating blow.

Gödel’s Incompleteness Theorem says that any system that is complex enough to express mathematics cannot prove, by itself, that everything it says is true. It will always rely on something outside the system that you have to assume is true but cannot prove.

You can then step outside the system and complete your proof, but in order to do that you will now have to invoke something else from the outside. So you keep expanding ever outward, invoking still more things that you cannot prove.

This was very disturbing to mathematicians, because mathematicians hate uncertainty.

Many people have raised the question of whether Gödel’s incompleteness theorem applies to the universe itself. If the universe is mathematical, then yes in fact it does.

Stated in Formal Language:

Gödel’s theorem says: “Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true, but not provable in the theory.”

The Church-Turing thesis says that a physical system can express elementary arithmetic just as a human can, and that the arithmetic of a Turing Machine (computer) is not provable within the system and is likewise subject to incompleteness.

Any physical system subjected to measurement is capable of expressing elementary arithmetic. (In other words, children can do math by counting their fingers, water flowing into a bucket does integration, and physical systems always give the right answer.)

Therefore the universe is capable of expressing elementary arithmetic and like both mathematics itself and a Turing machine, is incomplete.

Syllogism:

1. All non-trivial computational systems are incomplete

2. The universe is a non-trivial computational system

3. Therefore the universe is incomplete

Some time ago I posted an article about this: http://www.perrymarshall.com/articles/religion/godels-incompleteness-theorem/ and I was greatly interested in seeing if anyone would be able to poke a hole in my argument. (The article is a much more thorough explanation of Gödel than I am giving you here.)

Nearly everyone agrees that math is incomplete. The idea that the universe is also incomplete apparently makes some people very uncomfortable. If the universe cannot explain itself then there has to be some kind of higher power at work.

The debate essentially comes down to this:

If the universe is illogical and inconsistent then it is possible for it to be complete.
If the universe is logical and consistent then it is incomplete.
If the universe is incomplete, then it depends on something on the outside.

In other words, if the laws of mathematics and logic apply to the universe, then the universe has to have a metaphysical source. Atheism can only be true if the universe is irrational.

(By the way, my experience from conversing with literally thousands of atheists via email and on my various blogs is this: When you get down to the core emotional center of why they don’t believe in God, it’s often because they feel deep down that the universe is irrational. They’re immensely disappointed that the world is full of evil and suffering. Because of this, they reject the idea of God.)

You cannot prove that the universe is mathematical. But belief that the universe is mathematical is the #1 assumption of modern science. Toss out that assumption and the whole philosophical framework of western civilization crumbles.

In the history of science, you will find that belief in a God who created an orderly mathematical universe was one of the foundations of scientific discovery.

If you visit the world’s largest atheist website, Infidels, on the home page you will find the following statement:

“Naturalism is the hypothesis that the natural world is a closed system, which means that nothing that is not part of the natural world affects it.”

If you know Gödel’s theorem, you know that all logical systems must rely on something outside the system. So according to Gödel’s Incompleteness theorem, the Infidels cannot be correct. If the universe is logical, it has an outside cause.

Thus atheism violates the laws of reason and logic.

The Incompleteness of the universe isn’t formal proof that God exists. But… it IS proof that in order to construct a rational, scientific model of the universe, belief in God is not just 100% logical… it’s necessary.

Practically speaking, all knowledge we have about anything is incomplete. There are always some things you’re certain of, some things you’re somewhat sure of, and some things you cannot prove at all. Human knowledge is always enlarging the circle of what is known, but every question that we answer just provokes more questions. In real human experience, the quest to enlarge that circle never stops.

And I would submit to you that this is the essence of faith, as actually practiced by thinking, reasoning people.

Many people assume that religious faith is some mystical imaginary idea that is embraced purely on the basis of emotion or intuition. That it has nothing to do with facts, reason or logic.

This is completely untrue – at least in in the Judeo-Christian tradition. No one is asking you to believe without evidence or rational reason. Belief in God, in Jesus, and even the afterlife is based on historical events, logical propositions, and reasonable arguments.

Science itself originated from theology. Science assumed then, and assumes now, that the universe is rational. That the universe operates according to fixed, discoverable laws. Even science itself is a very practical outworking of faith in the reliability and consistency of the natural order.

The practice of faith is in many ways living out a hypothesis: That if you follow the teachings and embrace the Spirit, you will have an excellent opportunity to experience success in your work and your family. And that you will be rewarded in your search for meaning and pursuit of the deepest questions.

Perry Marshall

P.S.: If this intrigues you, make sure you read my more extended article, Gödel’s Incompleteness Theorem: The #1 Mathematical Discovery of the 20th Century

By: steven

steven — Wed, 24 Aug 2011 14:59:17 +0000

consider an opposite notion
can a state of nothing exist
can it be called a state
can it exist

By: Perry

Perry — Thu, 23 Jun 2011 00:52:23 +0000

Christian,

Thank you for the cordial debate. I see from a Google search that you have credentials in mathematics and it’s great to have this discussion with a qualified individual.

You said,

This need not be true: (a). That the universe is in reality an axiomatic mathematical system i.e. that everything in the universe we perceive as being real can be deduced mathematically from a set of axioms.

I am not saying that the universe ontologically IS an axiomatic mathematical system. I am saying that it strictly obeys mathematical laws and the laws of logic. Mathematical models of physical systems are isomorphic with those systems.

We have two mutually exclusive choices:

1. The universe does not obey mathematical laws.

2. The universe obeys mathematical laws.

If you choose (1) then the foundations of science go out the window.

If you choose (2) then all mathematical truths also apply to the universe. If 1+1=2 in math then 1+1 also equals 2 when you’re counting rocks. It makes no difference. Integration, multiplication, division are all the same with numbers as they are with physical objects, magnetic fields, energy, etc.

If we discover a new property of algebra tomorrow, we can be sure that it works in applied mathematics just as well as the algebra we already knew.

Syllogism:

1. All mathematical truths apply to physical systems
2. Incompleteness is a mathematical truth
3. Therefore incompleteness applies to physical systems

Statement 1 above is an axiom, it’s not provable. But it’s arguably the #1 axiom in all of science. It’s simply a statement that the universe is logical.

Now to your second statement:

(b). Even if (a) were true (which I have just shown you need not be the case at all), you also have to assume that the axioms of the universe form a computably enumerable set in order to apply Godel’s theorems.

Again we have two mutually exclusive choices:

1. The axioms of the universe are not computably enumerable

2. The axioms of the universe are computably enumerable

If you choose #1, that means the universe relies on either an infinite number of axioms, or else a set axioms which are illogical.

If the universe relies on illogical axioms then once again you’ve kicked the stool out from under science.

If the universe relies on an infinite number of axioms then that’s infinite regress. “Turtles all the way down.”

So our choices are:

1. The universe relies on an infinite set of axioms

2. The universe relies on a finite set of axioms.

#1 violates parsimony. So #2 is the only workable assumption. In mathematics you can make an infinite number of statements from a finite set of axioms but you can’t do anything with an infinite number of axioms.

The assumption that (a) the universe obeys mathematical laws and (b) it relies on a finite set of axioms is the only scenario that honors the principles of both science and math.

Perry

By: Christian Salas

Christian Salas — Tue, 21 Jun 2011 18:07:02 +0000

I’ll try again, being fully explicit this time, because with all due respect I think you have some confusions which prevent you from grasping what I said earlier. I will leave no room for doubt now (I hope). For Godel’s theorems to be applicable to the universe you have to assume two main things, which I will show need not be true at all:

(a). That the universe is in reality an axiomatic mathematical system i.e. that everything in the universe we perceive as being real can be deduced mathematically from a set of axioms.

There is no reason whatsoever why this should be true. In science and applied math what we are doing is using OUR OWN axiomatic mathematical systems to approximate certain features of the universe (e.g. by means of differential equations). Just because we are able to use our own axiomatic systems to do this, it does not mean that the whole universe itself must therefore be an axiomatic system! The latter does not follow logically from the former. It is a logical non sequitur.

So the first insurmountable weakness of your case is that there is no need whatsoever for the universe as a whole to be an axiomatic mathematical system. Science and math do NOT presume this. In doing science and math we are just using OUR OWN axiomatic systems to approximate certain aspects of the universe’s behaviour. That does not require the universe itself to be an axiomatic system.

And note again that the Church-Turing Thesis has NOTHING to do with this. All the Church-Turing thesis says is that algorithmically computable functions are recursive. There are many non-recursive functions which are not algorithmically computable. This has nothing whatsoever to do with the question of whether the universe itself is an axiomatic mathematical system. The two things are completely unrelated.

(b). Even if (a) were true (which I have just shown you need not be the case at all), you also have to assume that the axioms of the universe form a computably enumerable set in order to apply Godel’s theorems. What this means is that, in principle, a computer could list out all the universe’s axioms fully, even if this required the computer to carry on listing forever. In principle, even if it took forever, the computer could list out all the universe’s axioms. This is what it means to be computably enumerable, and this is what is required to apply Godel’s theorems.

Again, there is absolutely no need for this to be true for the axioms of the universe. There are many simple things that cannot be listed out fully by a computer even if it carried on forever e.g. the set of all real numbers. The set of all real numbers is said to be ‘uncountably infinite’ because no matter how hard you try to create a list of all real numbers, you will always find that you have missed out real numbers from the list. A computer could not do it any better than we could, even if it carried on forever trying to produce the list. The set of all real numbers is ‘un-listable’. This is what it means to be non-computably enumerable.

Now, there is no need whatsoever for the axioms of the universe to be computably enumerable. Even if we perceive everything in the universe as finite (which again is an open question in science – it is by no means certain), this does not mean that the universe’s AXIOMS have to be computably enumerable. The axioms could be an uncountably infinite set, and still give us a universe in which we perceive reality as being finite. You simply cannot assume from the perceived finiteness of the universe that the universe’s AXIOMS themselves are computably enumerable. That is again a logical non sequitur.

And again, science and math do not in any way presume that the AXIOMS of a mathematical universe are computably enumerable. All we need in order to do science and math is that the universe can be approximated by OUR OWN axiomatic systems. The fact that we can approximate certain features of the universe using our own axioms in no way implies that the axioms of the universe must themselves be computably enumerable.

In summary, there is absolutely no need whatsoever for assumptions (a) and (b) to be true. We can do science and math perfectly well without them, because all we need in order to do science and math is that OUR OWN axiomatic systems allow us to approximate some features of the universe e.g. using differential equations. The fact that we can do this in no way implies that the universe itself must satisfy the assumptions of Godel’s theorem.

I repeat again, your whole case is fundamentally flawed because you are assuming that (a) and (b) are true when there is no need for this to be the case. We do NOT need (a) and (b) to be true in order to use OUR OWN axiomaic systems in science and math to approximate some features of the universe. You DO need (a) and (b) in order to apply Godel’s theorems to the universe however.

By: Perry

Perry — Mon, 20 Jun 2011 23:30:01 +0000

There is no infinite set of real numbers in the universe, because the universe is finite. To the best of our actual knowledge, everything in the universe is finite. To the best of our knowledge, everything in the universe is also countable, measurable, weighable, quantifiable. Science always assumes this to be true.

A possible exception to this might be consciousness, in which case consciousness has a metaphysical component. Which brings us to the age-old “mind/body problem” and the question of free will, which I think we can set aside for the moment.

Outside of that, everything in the hard sciences is quantifiable and computable.

I reiterate that science and applied mathematics do presume computable enumerability of the universe. It’s not provable, though I certainly would love the fame and fortune that would come with proving it.

If you assume that it is not enumerable then there is nowhere left for science to go.

By: Christian Salas

Christian Salas — Mon, 20 Jun 2011 09:59:30 +0000

Your statement that science and math implicitly presume computable enumerability of the universe is completely false. It is not necessary for the universe to be computably enumerable in order to do science and math. To give a simple example, the set of all real numbers is an uncountably infinite set and therefore not computably enumerable. Despite this, we use real numbers constantly in science, math and everyday life. There is no need whatsoever for the universe to be computably enumerable in order to be able to do science and math in it. Science and math do not at all presume that the universe is computably enumerable.

You also seem to have a misunderstanding of the Church-Turing Thesis. The Church-Turing Thesis is a statement about the algorithmic computability of functions (it is an unproved hypothesis that all such functions are recursive). There are non-recursive functions (one famous one is called the Halting Problem) which are not algorithmically computable. Again, this has nothing whatsoever to do with the computable enumerability of the universe.

In order to apply Godel’s theorems to an axiomatic mathematical system you must first show that the axioms of that system are computably enumerable. As I said in my previous message and repeated above, there is absolutely no reason to think this is true for the universe (even if we accept that the universe is an axiomatic mathematical system – if you could prove that you would win the Nobel Prize in physics and you’d be more famous than Einstein). We could still do science and math perfectly well even if the axioms of the universe were not computably enumerable.

I am afraid your whole case is based on assumptions which are just as huge as the assumption “god exists”. You are essentially trying to prove one huge assumption (“god exists”) by implicitly making other huge assumptions (“the universe is an axiomatic mathematical system with computably enumerable axioms”) which are just as huge. There is no reason whatsoever why the latter should be true.

Your whole case is fundamentally flawed I’m afraid.