7 ms·

Gödel's ontological proof

55 points by kdavis 12y ago

michaelfeathers 12y ago
When I think of Godel, the first thing I think of (based on accounts) is compulsion. He was really driven to explore the edges of reality using logic as a probe. Here's an account of his discovery of a logical flaw in the US Constitution: http://morgenstern.jeffreykegler.com/ http://morgenstern.jeffreykegler.com/
- SilasX 12y ago
  Not really an account of his discovery, just a story about how he claimed to have found one; no one knows what flaw he thinks he found (but one can guess).
vilhelm_s 12y ago
Neel Krishnaswami wrote a quite readable description of the proof on his blog: http://semantic-domain.blogspot.com/2014/06/g-ontological-argument.html http://semantic-domain.blogspot.com/2014/06/g-ontological-ar...
michaelsbradley 12y ago
Readers might also enjoy the audio recording and transcript of a famous debate between Bertrand Russell and Fr. Frederick Copleston: The Famous 1948 BBC Radio Debate on the Existence of God http://www.biblicalcatholic.com/apologetics/p20.htm http://www.biblicalcatholic.com/apologetics/p20.htm
- jonahx 12y ago
  "Your general point, then, Lord Russell, is that it's illegitimate even to ask the question of the cause of the world" "Yes, that's my position" "Well, if it's a question that for you has no meaning, it's of course very difficult to discuss it, isn't it?" Russell, almost with a sigh: "Yes, it is very difficult." Then: "What do you say, shall we pass on to some other issue?" "Let's." What struck me most was the calmness and politesse they both maintained, and the steadfast articulateness at the speed of thought. There seemed to be an unspoken, shared assumption that you never, ever raise your voice or betray any hint of frustration, let alone insult your opponent in public. It's hard to imagine hearing something similar on the radio today. I'm also curious how much of the decorum was a sign of the times, and how much was specifically British.
  - philwelch 12y ago
    Russell and Copleston were titans of philosophy as well, and the philosophers I've seen tend to discuss philosophy in a calm, polite, and considered fashion.
lisper 12y ago
Godel's proof is essentially Anselm's with some mathematical window dressing added. It is worth noting that Anselm's argument applies equally well to Satan as to God: The most evil thing we can conceive of would be more evil if it actually existed, therefore Satan must actually exist. If you're into this sort of mind game, it makes an interesting exercise to translate this version of Anselm's argument into formal modal logic.
- michaelsbradley 12y ago
  Would it really apply to Satan? Historically, most Christian thinkers consider him to be a creature and thus finite in every respect; likewise, evil is generally considered by the same thinkers to be insubstantial, i.e. it is the absence of good and not some kind of entity (or put another way, it's not a matter–anti-matter kind of thing). So it seems evil would have a kind of local maximum, or floor value, depending on your perspective, which would be "zero good". At the same time we could speak of evil being compounded over time, i.e. an accumulation of depraved acts and persons. Of course, you may have other definitions of Satan and evil in mind, and I don't mean to imply that those concepts are solely the province of Christian minds.
  - lisper 12y ago
    One must not get too hung up on terminology here. The point is that the same argument that applies to "the greatest good that can be conceived" applies just as well to "the greatest evil that can be conceived." You can attach whatever labels to these things you like. The point is that if a great good becomes greater by actually existing, then so does a great evil, and so you can apply Anselm's rhetorical trick just as well to the latter as to the former.
    - michaelsbradley 12y ago
      But a "semantics of evil" was the point I was trying to bring to the fore. If the maximum local evil (e.g. in a person) is "zero good", i.e. the complete absence of good, then the greatest evil is a fixed value and doesn't seem subject to the same line of reasoning.
riffraff 12y ago
and an implementation (I think unfinished) in Coq https://github.com/FormalTheology/GoedelGod https://github.com/FormalTheology/GoedelGod
amelius 12y ago
It seems to me that he left out a couple of "if it is possible to exist" clauses here and there.
- juped 12y ago
  This is exactly the opposite of what is the case. Anselm implicitly assumed that it was possible for God to exist, but Gödel proves it as a theorem (theorem 2 on the Wikipedia page).
KierkegaardE 12y ago
This proof is a mix between Anselm and a little bit of math. Disclaimer; I've only read Anselm and Leibniz, and am just trying to piece together the discrepant sources around. Premise 1: There are many worlds. Definition 1: "x necessarily exists if and only if every essence of x is necessarily exemplified" (From the article) Premise 2: Existence is good. Definition 2: God is the being Good-er than whom cannot be conceived. Sub-Conclusion 1: God if he exists, by definition is good, and by supposition necessarily exists because existence is good. He necessarily exists because if he did not, then a being good-er than him could be conceived (ie, one that existed), and that being would then not be god. sub-conclusion 2: Because there are many worlds, there exists one with a being good-er than whom cannot be conceived. Therefore, this is a being whose existence is necessary. Consider then, that God's existence would not be necessary, if the world that god governed was not necessary, because that world could, or could not exist, and so God could, or could not exist, and god's existence would not be necessarily be exemplified. Therefore The world in which God exists must be necessary to our world, and by consequence, God's existence must be necessary to our world. If someone who has actually read him could give me some feedback as to whether this is fairly close to what he means, I'd appreciate it.
- s_baby 12y ago
  >Premise 1: There are many worlds. The premise is called the plenitude principle. "The principle of plenitude asserts that the universe contains all possible forms of existence."[1] You first see it in Plato's work like in the Cave Allegory. You can make a compelling argument that the plenitude principle was the precursor to the ideas of a multiverse and evolution. 1. http://en.wikipedia.org/wiki/Principle_of_plenitude http://en.wikipedia.org/wiki/Principle_of_plenitude
gjm11 12y ago
The difference between Anselm's original ontological argument and modern ones like Goedel's or Plantinga's is that while Anselm's is full of terrible reasoning, Goedel's reasoning is (unsurprisingly) watertight and all the terribleness is concentrated in the axioms. Specifically, I think the idea of classifying all properties as "good" or "not good" is hopeless: I do not believe there is any such classification that fits anyone's intuitive ideas about goodness well enough that calling something "godlike" if it has all "good" properties is credible. More specifically, I suspect it's consistent to suppose -- hence there are no humanly-comprehensible counterexamples -- that the only notions of goodness obeying all Goedel's axioms are ones that look like "P is good iff P(x)" for some fixed object x. (And, like Krishnaswami, I think the system of modal logic Goedel needs is awfully strong.) In the spirit of Anselm's ontological argument, however, I offer the following proof of the nonexistence of God: Consider a really bad argument for the existence of God. In fact, consider one so bad that no worse argument can be conceived. Obviously a bad argument for something fails to prove what it purports to prove. But merely failing is a pretty mediocre kind of badness. The worst possible argument, surely, has to be much worse than that: it must conclusively prove the opposite of what it's meant to prove. Now, the worst conceivable argument for theism clearly "exists in the understanding", as St Anselm put it. But it can't exist only there -- because a bad argument is more damaging to the premise it's meant to support if it's actually made. Therefore, there is an argument for the existence of God which is actually a conclusive proof of the nonexistence of God. And, of course, any proposition that can be conclusively disproved is false; therefore there is no God. (This argument is in my opinion almost exactly as strong as the original ontological argument for the existence of God. Which is to say, it's absolutely hopeless. But I think it's fun.)
- diminish 12y ago
  Yet another interesting side of Anselm's and Gödel's proof is that they leave the possibility open for us to be gods.
- logicallee 12y ago
  wow! did you come up with this?
tunesmith 12y ago
I only dabble in formal logic so I haven't come across this before, but it sort of feels like this rests on a circular argument. You know, the "can God make a rock so heavy that he can't lift it" or "does God have the power to make himself even more Good". I wonder if it's possible to prove anything if one of your axioms is actually a hidden circular argument, much like how one can prove anything from setting true to false.
- davidgerard 12y ago
  All ontological arguments for God are of the form: 1. I carefully define God as having to exist. 2. Therefore, God exists. All the work goes into the first part, however many volumes of obfuscation (hi, Dr Plantinga!) one hides it behind. There are a number of problems with this approach.
- dvt 12y ago
  What you're describing in the second part of your post is the Principle of Explosion[1]. It does not apply to Godel's argument as his axioms entail no contradictions. I'm not sure what you mean by "circular" in this context. However, what people usually mean is an argument where the conclusion is contained in one of the axioms. Godel's proof doesn't explicitly suffer from this but I will say that his fourth axiom, i.e. Good(phi) -> [Necessarily]Good(phi) is cutting it pretty close. [1] http://en.wikipedia.org/wiki/Principle_of_explosion http://en.wikipedia.org/wiki/Principle_of_explosion
eli_gottlieb 12y ago
This assumes that "goodness" is an irreducible, ontologically basic property.
chroma 12y ago
Note that even if you accept this proof, it says nothing about a god that answers prayers or otherwise intervenes in daily life. It also says nothing about an afterlife. Unless you're a deist, ontological arguments are practically useless. They've been used in the defense of many gods now relegated to mythology.
- elwell 12y ago
  Or the FSM
- gjm11 12y ago
  I don't know about that. If you accept an ontological argument as valid, what it gives you is a god who's maximally perfect in every way. So then if you want to show (e.g.) that he answers prayers, all you have to do is to argue that answering prayers is more perfect than ignoring them. Descartes gives a kinda-ontological argument for the existence of God, and then he does indeed very rapidly move to deducing useful properties of God from it -- e.g., that he wouldn't allow us to be systematically deceived about everything. It's the basis of his whole epistemology, at least ostensibly. Whereas, say, a first-cause argument tells you there's something that's in some sense the cause or origin of the universe, but you've got a whole lot of work to do to get from there to anything relevant to actual religious belief. (Though, e.g., William Lane Craig does somehow manage to keep a straight face when transitioning at lightning speed from "something caused the universe to begin to exist" to "and that something must be personal, enormously powerful, etc.) (Lest I be misunderstood, I'll add that I think both ontological and cosmological arguments fail badly, and that neither of them either manages to give much reason to believe in a maximally perfect being / first cause / whatever, or to believe that such a being if it existed would have anything much to do with the gods of traditional religions. But it's not my purpose to litigate any of those claims here -- Goedel's argument is much more interesting than any I'd make.)
  - chroma 12y ago
    I thought the same thing as you (answers prayers == more perfect), but figured proponents of the argument would ignore that. If a perfect being must answer prayers and prayers aren't answered (which seems to be the case), then it's kind of hard for a perfect being to exist. Either that or everyone is praying to the wrong gods.
elwell 12y ago
We're still talking about God? There must be something to this "God" idea that is important / common to humanity.
- sjwright 12y ago
  The gullibility of children, usefully instilled into us by natural selection. The fear of death, usefully instilled into us by natural selection. Human minds have no experience of not existing, and tend to find the concept difficult to comprehend. We desire answers, but unfortunately the human condition is to prefer wrong answers over no answer.
  - amelius 12y ago
    Human brains also deceive us into believing we are distinct entities, whereas in reality the universe is probably a single entity.
    - sjwright 12y ago
      Cute, and in some respects insightful, but as a contribution it's arguably an equivocation on the definition of entity. You could similarly argue that human brains deceive us into believing we are distinct entities, whereas in reality we are billions of entities experiencing the emergent properties of multi-cellular life.
    - elwell 12y ago
      > probably a single entity "Probably"? How about "surely" or "surely not"? Is an entity comprised of many things or one thing? Does it matter? Are you merely pointing out that there is no true separation between anything and everything that exists? http://en.wikipedia.org/wiki/Monism http://en.wikipedia.org/wiki/Monism
- deleted 12y ago
  [deleted]
sjwright 12y ago
We see our world in terms of cause and effect, but these are temporal concepts that might have no meaning outside of linear time. It may therefore be nonsensical to ask what caused the universe.

55 points by kdavis 12y ago

When I think of Godel, the first thing I think of (based on accounts) is compulsion. He was really driven to explore the edges of reality using logic as a probe. Here's an account of his discovery of a logical flaw in the US Constitution: http://morgenstern.jeffreykegler.com/ http://morgenstern.jeffreykegler.com/

SilasX 12y ago
Not really an account of his discovery, just a story about how he claimed to have found one; no one knows what flaw he thinks he found (but one can guess).

vilhelm_s 12y ago

Neel Krishnaswami wrote a quite readable description of the proof on his blog: http://semantic-domain.blogspot.com/2014/06/g-ontological-argument.html http://semantic-domain.blogspot.com/2014/06/g-ontological-ar...

michaelsbradley 12y ago

Readers might also enjoy the audio recording and transcript of a famous debate between Bertrand Russell and Fr. Frederick Copleston: The Famous 1948 BBC Radio Debate on the Existence of God http://www.biblicalcatholic.com/apologetics/p20.htm http://www.biblicalcatholic.com/apologetics/p20.htm

jonahx 12y ago
"Your general point, then, Lord Russell, is that it's illegitimate even to ask the question of the cause of the world" "Yes, that's my position" "Well, if it's a question that for you has no meaning, it's of course very difficult to discuss it, isn't it?" Russell, almost with a sigh: "Yes, it is very difficult." Then: "What do you say, shall we pass on to some other issue?" "Let's." What struck me most was the calmness and politesse they both maintained, and the steadfast articulateness at the speed of thought. There seemed to be an unspoken, shared assumption that you never, ever raise your voice or betray any hint of frustration, let alone insult your opponent in public. It's hard to imagine hearing something similar on the radio today. I'm also curious how much of the decorum was a sign of the times, and how much was specifically British.
- philwelch 12y ago
  Russell and Copleston were titans of philosophy as well, and the philosophers I've seen tend to discuss philosophy in a calm, polite, and considered fashion.

lisper 12y ago

Godel's proof is essentially Anselm's with some mathematical window dressing added. It is worth noting that Anselm's argument applies equally well to Satan as to God: The most evil thing we can conceive of would be more evil if it actually existed, therefore Satan must actually exist. If you're into this sort of mind game, it makes an interesting exercise to translate this version of Anselm's argument into formal modal logic.

michaelsbradley 12y ago
Would it really apply to Satan? Historically, most Christian thinkers consider him to be a creature and thus finite in every respect; likewise, evil is generally considered by the same thinkers to be insubstantial, i.e. it is the absence of good and not some kind of entity (or put another way, it's not a matter–anti-matter kind of thing). So it seems evil would have a kind of local maximum, or floor value, depending on your perspective, which would be "zero good". At the same time we could speak of evil being compounded over time, i.e. an accumulation of depraved acts and persons. Of course, you may have other definitions of Satan and evil in mind, and I don't mean to imply that those concepts are solely the province of Christian minds.
- lisper 12y ago
  One must not get too hung up on terminology here. The point is that the same argument that applies to "the greatest good that can be conceived" applies just as well to "the greatest evil that can be conceived." You can attach whatever labels to these things you like. The point is that if a great good becomes greater by actually existing, then so does a great evil, and so you can apply Anselm's rhetorical trick just as well to the latter as to the former.
  - michaelsbradley 12y ago
    But a "semantics of evil" was the point I was trying to bring to the fore. If the maximum local evil (e.g. in a person) is "zero good", i.e. the complete absence of good, then the greatest evil is a fixed value and doesn't seem subject to the same line of reasoning.

riffraff 12y ago

and an implementation (I think unfinished) in Coq https://github.com/FormalTheology/GoedelGod https://github.com/FormalTheology/GoedelGod

amelius 12y ago

It seems to me that he left out a couple of "if it is possible to exist" clauses here and there.

juped 12y ago
This is exactly the opposite of what is the case. Anselm implicitly assumed that it was possible for God to exist, but Gödel proves it as a theorem (theorem 2 on the Wikipedia page).

KierkegaardE 12y ago

This proof is a mix between Anselm and a little bit of math. Disclaimer; I've only read Anselm and Leibniz, and am just trying to piece together the discrepant sources around. Premise 1: There are many worlds. Definition 1: "x necessarily exists if and only if every essence of x is necessarily exemplified" (From the article) Premise 2: Existence is good. Definition 2: God is the being Good-er than whom cannot be conceived. Sub-Conclusion 1: God if he exists, by definition is good, and by supposition necessarily exists because existence is good. He necessarily exists because if he did not, then a being good-er than him could be conceived (ie, one that existed), and that being would then not be god. sub-conclusion 2: Because there are many worlds, there exists one with a being good-er than whom cannot be conceived. Therefore, this is a being whose existence is necessary. Consider then, that God's existence would not be necessary, if the world that god governed was not necessary, because that world could, or could not exist, and so God could, or could not exist, and god's existence would not be necessarily be exemplified. Therefore The world in which God exists must be necessary to our world, and by consequence, God's existence must be necessary to our world. If someone who has actually read him could give me some feedback as to whether this is fairly close to what he means, I'd appreciate it.

s_baby 12y ago
>Premise 1: There are many worlds. The premise is called the plenitude principle. "The principle of plenitude asserts that the universe contains all possible forms of existence."[1] You first see it in Plato's work like in the Cave Allegory. You can make a compelling argument that the plenitude principle was the precursor to the ideas of a multiverse and evolution. 1. http://en.wikipedia.org/wiki/Principle_of_plenitude http://en.wikipedia.org/wiki/Principle_of_plenitude

gjm11 12y ago

The difference between Anselm's original ontological argument and modern ones like Goedel's or Plantinga's is that while Anselm's is full of terrible reasoning, Goedel's reasoning is (unsurprisingly) watertight and all the terribleness is concentrated in the axioms. Specifically, I think the idea of classifying all properties as "good" or "not good" is hopeless: I do not believe there is any such classification that fits anyone's intuitive ideas about goodness well enough that calling something "godlike" if it has all "good" properties is credible. More specifically, I suspect it's consistent to suppose -- hence there are no humanly-comprehensible counterexamples -- that the only notions of goodness obeying all Goedel's axioms are ones that look like "P is good iff P(x)" for some fixed object x. (And, like Krishnaswami, I think the system of modal logic Goedel needs is awfully strong.) In the spirit of Anselm's ontological argument, however, I offer the following proof of the nonexistence of God: Consider a really bad argument for the existence of God. In fact, consider one so bad that no worse argument can be conceived. Obviously a bad argument for something fails to prove what it purports to prove. But merely failing is a pretty mediocre kind of badness. The worst possible argument, surely, has to be much worse than that: it must conclusively prove the opposite of what it's meant to prove. Now, the worst conceivable argument for theism clearly "exists in the understanding", as St Anselm put it. But it can't exist only there -- because a bad argument is more damaging to the premise it's meant to support if it's actually made. Therefore, there is an argument for the existence of God which is actually a conclusive proof of the nonexistence of God. And, of course, any proposition that can be conclusively disproved is false; therefore there is no God. (This argument is in my opinion almost exactly as strong as the original ontological argument for the existence of God. Which is to say, it's absolutely hopeless. But I think it's fun.)

diminish 12y ago
Yet another interesting side of Anselm's and Gödel's proof is that they leave the possibility open for us to be gods.
logicallee 12y ago
wow! did you come up with this?

tunesmith 12y ago

I only dabble in formal logic so I haven't come across this before, but it sort of feels like this rests on a circular argument. You know, the "can God make a rock so heavy that he can't lift it" or "does God have the power to make himself even more Good". I wonder if it's possible to prove anything if one of your axioms is actually a hidden circular argument, much like how one can prove anything from setting true to false.

davidgerard 12y ago
All ontological arguments for God are of the form: 1. I carefully define God as having to exist. 2. Therefore, God exists. All the work goes into the first part, however many volumes of obfuscation (hi, Dr Plantinga!) one hides it behind. There are a number of problems with this approach.
dvt 12y ago
What you're describing in the second part of your post is the Principle of Explosion[1]. It does not apply to Godel's argument as his axioms entail no contradictions. I'm not sure what you mean by "circular" in this context. However, what people usually mean is an argument where the conclusion is contained in one of the axioms. Godel's proof doesn't explicitly suffer from this but I will say that his fourth axiom, i.e. Good(phi) -> [Necessarily]Good(phi) is cutting it pretty close. [1] http://en.wikipedia.org/wiki/Principle_of_explosion http://en.wikipedia.org/wiki/Principle_of_explosion

eli_gottlieb 12y ago

This assumes that "goodness" is an irreducible, ontologically basic property.

chroma 12y ago

Note that even if you accept this proof, it says nothing about a god that answers prayers or otherwise intervenes in daily life. It also says nothing about an afterlife. Unless you're a deist, ontological arguments are practically useless. They've been used in the defense of many gods now relegated to mythology.

elwell 12y ago
Or the FSM
gjm11 12y ago
I don't know about that. If you accept an ontological argument as valid, what it gives you is a god who's maximally perfect in every way. So then if you want to show (e.g.) that he answers prayers, all you have to do is to argue that answering prayers is more perfect than ignoring them. Descartes gives a kinda-ontological argument for the existence of God, and then he does indeed very rapidly move to deducing useful properties of God from it -- e.g., that he wouldn't allow us to be systematically deceived about everything. It's the basis of his whole epistemology, at least ostensibly. Whereas, say, a first-cause argument tells you there's something that's in some sense the cause or origin of the universe, but you've got a whole lot of work to do to get from there to anything relevant to actual religious belief. (Though, e.g., William Lane Craig does somehow manage to keep a straight face when transitioning at lightning speed from "something caused the universe to begin to exist" to "and that something must be personal, enormously powerful, etc.) (Lest I be misunderstood, I'll add that I think both ontological and cosmological arguments fail badly, and that neither of them either manages to give much reason to believe in a maximally perfect being / first cause / whatever, or to believe that such a being if it existed would have anything much to do with the gods of traditional religions. But it's not my purpose to litigate any of those claims here -- Goedel's argument is much more interesting than any I'd make.)
- chroma 12y ago
  I thought the same thing as you (answers prayers == more perfect), but figured proponents of the argument would ignore that. If a perfect being must answer prayers and prayers aren't answered (which seems to be the case), then it's kind of hard for a perfect being to exist. Either that or everyone is praying to the wrong gods.

elwell 12y ago

We're still talking about God? There must be something to this "God" idea that is important / common to humanity.

sjwright 12y ago
The gullibility of children, usefully instilled into us by natural selection. The fear of death, usefully instilled into us by natural selection. Human minds have no experience of not existing, and tend to find the concept difficult to comprehend. We desire answers, but unfortunately the human condition is to prefer wrong answers over no answer.
- amelius 12y ago
  Human brains also deceive us into believing we are distinct entities, whereas in reality the universe is probably a single entity.
  - sjwright 12y ago
    Cute, and in some respects insightful, but as a contribution it's arguably an equivocation on the definition of entity. You could similarly argue that human brains deceive us into believing we are distinct entities, whereas in reality we are billions of entities experiencing the emergent properties of multi-cellular life.
  - elwell 12y ago
    > probably a single entity "Probably"? How about "surely" or "surely not"? Is an entity comprised of many things or one thing? Does it matter? Are you merely pointing out that there is no true separation between anything and everything that exists? http://en.wikipedia.org/wiki/Monism http://en.wikipedia.org/wiki/Monism
deleted 12y ago
[deleted]

sjwright 12y ago

We see our world in terms of cause and effect, but these are temporal concepts that might have no meaning outside of linear time. It may therefore be nonsensical to ask what caused the universe.