Today in 1978, the logician Kurt Gödel died in Princeton, New Jersey. Gödel, of course, is remembered for his incompleteness theorems but also took the ontological proof for the existence of God serious enough to express his own version of it in modal logic.
Strangely, Richard Dawkins does not mention Gödel's version in The God Delusion, and instead restricts himself to discussing Anslem's version presented in Proslogium (over 900 years earlier) which Dawkins describes as "infantile". For that matter, he also doesn't mention versions of the ontological argument developed by, for example, Descartes, Leibniz or Plantinga. If anything is "infantile"
it is Dawkins' treatment of the argument - for a more mature work, try The Cambridge Companion to Atheism.
Refs:
Kurt Gödel (1995) "Ontological Proof". Collected Works: Unpublished Essays & Lectures, Volume III. (Oxford, pp. 403-404).
Kenneth Einar Himma (2006) "Ontological Argument" Internet Encyclopedia of Philosophy.
Graham Oppy (2005) "Ontological Arguments" Stanford Encyclopedia of Philosophy.
- Log in to post comments
On the other hand, Godel didn't have a website.
It's the problem with infinites. Infinity is a very strange number in math, and in logic.
People really don't understand them. They put a very heavy thumb on one side of the equation.
What's next, Bayesian probability of God's existence?
I do think it's somewhat infantile to expect that one can define God into existence by whatever structure of human logic you want.
Especially when nobody seems to understand infinity. They seem to think they can lump any old infinite thing they want into that infinity.
I'm not saying I'm a more brilliant logician than Godel. What I'm saying is that logic is a method invented by man to try and understand the world. It's only as good as the brains that built it, and, you know, garbage in-garbage out. It's useful within its own rules, but it cannot be a primary research tool. To use it to define God is to declare logic infinite and complete.
Two things that Godel told us quite a bit about.
Goedel's proof of the existence of god is, to put it mildly, a heap of rotting dodo dung. Just because he was a brilliant meta-logician does not mean that when it came to religion he was any more rational than anybody else.
Hardly a proof of its inadequacy, now is that?
Anyway, that misses my point totally ...
I keep seeing the ontological arguments for the existence of god discussed on various blogs here at ScienceBlogs and elsewhere, and I really don't get it. At its core is the (what should be obvious) fact that if you start with a false assumption, you can logically derive any conclusion you like. Recasting such an argument using more sophisticated mathematical tools such as modal logic does not change that basic fact.
I was under the (evidently mistaken) impression that the ontological argument for god was mainly of historical interest. And yet I keep seeing people discussing these arguments as if there is some serious content there. (And since some, like Chris, identify themselves as atheists, wouldn't they therefore necessarily believe the arguments are false?)
I just don't get it.
Why should Dawkins have treated Gödels proof in stead of Anselms? Isn't Anselm the more known proof?
You seem to expect Dawkins to mention all versions of the ontological argument aver stated? Or just the ones you believe are good?
Do you have any specific complaints about Dawkins treatment of the ontological proof, other than the fact that he only looked at one of them and not all of them?
Like Kurt says it is really easy to dismiss such proofs, just deny the assumptions, and you are home safe.
Gödels proof is destrpyed if you do not accept that it is possible to single out positive properties from other properties, or deny that if P1 and P2 are positive by themselves then the vonjugation of them must be positive, or just to claim that P and non P can both be positive etc.
Or just deny the key assumption in both Anselm and Gödels proofs. That necessary existance is a positive property.
In fact denying this would dismiss both proofs wouldn't it? didn't Dawkins dismiss this assumption in his book? Wouldn't it be redundant of him to dismiss Anselms proof and then mention another with the same fatal flaw?
And please, have you read Dawkins book? It is not a learned treatise, its not meant as the ultimate word in all matters of faith, but it DOES present an outline of the most common types of god proofs and it outlines where they fail.
Dawkins references other works going into more datail on the proofs. An come on, doesn't Gödel, Anselms and Plantingas "proofs" sem very contrived to you?
You seem to expect Dawkins to mention all versions of the ontological argument aver stated? Or just the ones you believe are good?
I don't expect him to deal with all versions "avar stated", and don't find any of them convincing myself. I do however, expect Dawkins as an intellectual to properly engage with the arguments even in a popular work.
Like Kurt says it is really easy to dismiss such proofs, just deny the assumptions, and you are home safe.
Ahem. Shouldn't you as a rationalist be disproving the assumptions and not just denying them. Whether Gödel's proof may be "destrpyed" by denying that "existance is a positive property", surely you must prove your stance rather than just deny Gödel's
didn't Dawkins dismiss this assumption in his book?
Precisely. Dismiss is the key word here. Dawkins doesn't so much argue against Anselm as sneer at him. Dawkins "devotes" six pages to the ontological proof. Does he actually engage with the argument by trying (for example) to argue against its assumptions? No. Instead we are told that the argument is "infantile," we are given it in the "appropriate ... language of the playground", and we are told that it is "logomachist trickery" that offends Dawkins. When we actually get to a "disproof" (p. 83) we are presented with Norman Malcolm's statement regarding the queerness of the statemnt that existence is a perfection but are never told why we should accept Malcolm's view over that of Descartes, Leibniz or Gödel (who of course, don't get mentioned). Then we are treated to Douglas Gaskins' "funny" proof that God doesn't exist (which, by the way, is clearly logically problematic).
And please, have you read Dawkins book? It is not a learned treatise,
Yes I have. Damned straight it is not a "learned treatise". Just because a work is popular doesn't mean it cannot fully engage the opposing viewpoint or have to "dumb-down" the material. See, for example, Gleick's Chaos or Dawkins' own The Selfish Gene. Both deal with complicated material in a clear, concise fashion, and both are intellectually more respectable that Dawkins' current jermiad.
An come on, doesn't Gödel, Anselms and Plantingas "proofs" sem very contrived to you?
Just because they "sem very contrived" doesn't mean that they can be dismissed quickly.
As far as I can tell from Goedel's ontological argument, it's simply a modal formalization of Anslem's earlier work. I don't see why Dawkins should have dealt with it instead of the original argument by Anslem, considering that the use of mathematical esoterica is a bit extraneous.
What I especially don't get is why this is actually considered an argument:
"Now we define a new property G: if x is an object in some possible world, then G(x) is true if and only if P(x) is true in that same world for all positive properties P."
The whole assumption deals with treating syntactic contrivances as real entities. A positive property is an abstract construction, with no description of any physical realization. You can define anything into existence in this fashion. I don't see why Dawkins "sneering" fails to deal with this, he basically said what I just did with more humor.
As far as I can tell from Goedel's ontological argument, it's simply a modal formalization of Anslem's earlier work.
Actually (and I'm no expert on this obviously), it is a formalization of Liebniz's version (which differs from Anslem's).
I don't see why Dawkins "sneering" fails to deal with this, he basically said what I just did with more humor.
Depends on how you define "humor", I guess.
Godel had a pathological fear of being poisoned. He refused to eat any food that was not prepared by his wife. When she was hospitalized, he starved to death (apparently he didn't trust his own cooking). He used to open the window in the middle of winter for fear of a poison gas attack. Of course, he was also a brilliant logician. But his ontological proof is more in keeping with his loony side than with his brilliant side. There's a reason people don't mention it in the same breath with his famous proofs. Read it through if you understand it. Which, by the way, I doubt you do... which should give you your answer as to why Dawkins didn't discuss this "proof" in his book: his book is aimed at lay readers such as yourself.
Anyway, for those of you who think arguments like those of Anselm and Godel are actually worth considering, here's a proof of God's non-existence, formulated by Douglas Gasking:
1. The creation of the world is the most marvelous achievement imaginable.
2. The merit of an achievement is the product of (a) its intrinsic quality, and (b) the ability of its creator.
3.The greater the disability (or handicap) of the creator, the more impressive the achievement.
4.The most formidable handicap for a creator would be non-existence.
5. Therefore if we suppose that the universe is the product of an existent creator we can conceive a greater being -- namely, one who created everything while not existing.
6. Therefore God does not exist.
The point of my somewhat extreme comment is that Goedel's supposed proof is so "bad" that it would be a total waste of time and energy for Dawkins to discuss it. Even Hao Wang the first person to make it public after Goedel's death refused to discuss it! I think for the obvious reason that it is not worth discussing. In fact all the ontological proofs that have been presented are from a formal logical point of view, as Dawkins apparently said (I haven't read the original myself, childish and I don't understand why so many people are so upset at Dawkins for not wasting his time on them.
As I wrote my last post I had overseen your answer to Soren which actual answers my second posting from your standpoint sorry! However you write
"Ahem. Shouldn't you as a rationalist be disproving the assumptions and not just denying them. Whether Gödel's proof may be "destrpyed" by denying that "existance is a positive property", surely you must prove your stance rather than just deny Gödel's
Logically you can not disprove an assumption you can either accept or reject it that is why Aristotle required axioms, the assumptions of of formal deductive "scientia", should be of an obvious nature. The problem with the ontological arguments for the existence of a god is that they are circular. They all assume the existence of some abstract property then claim that god is this property and then conclude with the statement that god must therfore exist. This is just the same as saying god exists without the hand waiving of the so called proof.
[we'll try this again ... my first reply got swallowed by the aether]
Godel had a pathological fear of being poisoned. He refused to eat any food that was not prepared by his wife. When she was hospitalized, he starved to death (apparently he didn't trust his own cooking). He used to open the window in the middle of winter for fear of a poison gas attack. Of course, he was also a brilliant logician. But his ontological proof is more in keeping with his loony side than with his brilliant side.
We'll ignore the ad hominem and move on to ...
There's a reason people don't mention it in the same breath with his famous proofs. Read it through if you understand it. Which, by the way, I doubt you do...
I freely admit to not understanding Godel's argument - I was merely using it to make a broader point about Dawkins' willingness to caricature the ontological argument. But since you imply that you do undertand Godel, please take the time to enlighten me as to (a) what the argument is, (b) what's clearly wrong with it and (c) why philosophers like AP Hazen continue to write on it. Otherwise, you're just blowing smoke.
his book is aimed at lay readers such as yourself.
Yes, yes, yes. I'm well aware that Dawkins is writing for "lay readers" (which I assume you are as well since Dawkins is not writing within any recognizable academic field). However, that does not excise Dawkins from parodying and caricaturing arguments. I'm sick of making this point. It seems that some supporters of Dawkins just refuse to see that he is not fairly representing arguments for the existence of God (none of which, by the way, I find convincing).
[H]ere's a proof of God's non-existence, formulated by Douglas Gasking:
Nice job parroting Dawkins. Now what exactly does this argument show? That ontological arguments are incorrect? That God doesn't exist? Perhaps there are problems with Gasking's statements (hint, look at #5).
I've wasted enough time on this over the past month. I fail to see how anyone can find what Dawkins writes in Chapter 3 to be intellectually satisfying. If you do, it's likely that you have preconceptions that are being supported by Dawkins' glib "arguments". The book would have been far better without that chapter.
And that is my last word on The God Delusion ... you can keep on hammering away.
As the kids say, "lol"
Thanks for the explanation of why Godel is wrong. It was a pleasure to be enlightened by you.
Take a course in logic. I'm not in the business of explaining people's blog posts to them.
Ah now I see. You cant explain why Godel is wrong because (as you say here) you are a "a lowly high school student" and thus blowing smoke. Thanks for playing. Come back when you have actually read something substantial on the ontological argument. And that excludes Dawkins.
Axiom 5 in Godel's proof, "Necessary existence is a positive property," looks like it can be attacked by Kant's objection that "existence is not a predicate." Am I right?
BTW, I remember you mentioning in the comments of your post on "Dawkins' 'stock reply'," that you were going to critique Chapter 3 of TGD, though that post seems to have disappeared. Are you still planning on doing that (in your copious spare time, of course :-))?
you were going to critique Chapter 3 of TGD
I started to write something and realized that I was spending way to much time on TGD, and the effort was probably wasted all things considered. I'll stand by my claim that Chapter 3 greatly weakens the overall work and that it does not reflect anything interesting that's happening in philosophy of religion.
The only reason I mentioned the OA again was the Godel connection. Probably should have resisted the temptation!
And that is definitely my last word on the issue.
John Lynch: "I fail to see how anyone can find what Dawkins writes in Chapter 3 to be intellectually satisfying."
I suspect that your reaction to Chapter 3 is similar to that of a guy who reads an article in the newspaper about a topic with which he is familiar, and he sees all the places where the reporter screwed up. Someone less familiar with the material wouldn't notice the mistakes.
For those of you (seemingly only Dr. Lynch in this comments thread) who are interested in rigorous arguments dealing with the Ontological argument (Dawkins, unfortunately, does not do it), I suggest Alvin Plantinga's _Nature of Necessity_, and _God, Freedom, and Evil_. Howard Sobel's book _Logic and Thesism_ actually considers the six most famous ontological arguments (included Gödel's) in detail.
Also, if you are interested in Gödel the man, both Rebecca Goldstein's _Incompleteness_ and Palle Yourgrau's _A World Without Time_ are excellent.
Finally, w/r/t to Dawkins's 3rd chapter: it only gives 1) Dawkins, 2) 'pop' science/philosophy writers, and 3) atheists a bad name when 'public' intellectuals give half-baked arguments against serious philosophical positions. It is also a waste of the reader's time.
So all I have to do is learn modal logic, read that proof, and I will go, "A-ha! God exists!"?
So many people have told me that the ontological proofs are so clever and deep, but every time I've read the ones written in languages I can comprehend, they sound painfully vacuous. Are you really trying to tell me that this argument, written in a language I can't understand, is wonderfully powerful and persuasive?
Do you believe in a god?
I'm not sure that PZ is addressing me or perhaps Errol, but nonetheless ...
Are you really trying to tell me that this argument, written in a language I can't understand, is wonderfully powerful and persuasive?
No, just that chapter 3 of TGD is flawed and probably diminishes the worth of the book as a whole.
Do you believe in a god?
No.
PZ Meyers: "So many people have told me that the ontological proofs are so clever and deep"
Who are these people? Or are you confusing them with those who simply think that ontological proofs are not trivial to refute? Considering that refutations of ontological proofs hang on such subtle things as existence not being a property of an object, I'd say that the ontological arguments are in some sense clever.
Wow, searing critique of a poster's typos, Lynch, my man.
I myself stopped critiquing people's typos when I learned that handicapped people use the internet, and sometimes have physical trouble typing. I recommend it.
Ironically, Errol has posted a perfect example of the courtier's reply: you shouldn't diss Goedel or Platinga because hey, they're serious. But theology, regardless of how seriously you take it is like Oakland: There's no there there.
I'm with Kurt on this. Does anyone actually believe in god because of the ontological argument? Heck, do most people even know the ontological argument? Okay, Dawkins fails to present the known refutations of it; that's a valid criticism of his book. But it's hardly a valid criticism of his position.
I'm also with PZ in finding the OA vacuous. Its conclusion is that god's existence is necessary, which is clearly absurd. I can imagine a universe in which god does not exist quite easily. More generally, the notion that we can deduce deep truths about the world merely through the application of logic is quite strange to me, and I'm not sure why anyone would take it seriously.
The courtiers response gets really tedious fast.
I would think the point is (perhaps also Dawkins) that the ontological argument and other theological arguments are no longer interesting since they have been analyzed to death. Nothing has come out of that kind of millennium old reasoning.
Sure. Here is a sketch of one showing why gods are improbable:
P(N|D) < P(N|D&T); N = Natural universe, D = Data, T = Theory. Here each time we explain a data set by a natural theory the probability for the universe being natural increases.
No numbers for doing a hypothesis test, unfortunately. :-)
Wait a minute! WAIT A MINUTE!!!!
John Lynch: "Ahem. Shouldn't you as a rationalist be disproving the assumptions and not just denying them. Whether Gödel's proof may be "destrpyed" by denying that "existance is a positive property", surely you must prove your stance rather than just deny Gödel's "
That's *NOT* how it works. Anyone proposing a proof is free to start with any assumptions they want, but any resulting proof (assuming it is logically sound) is contingent on proof of the assumptions.
(I can conjur up some really wild conclusions though some logically concrete proofs based on the assumption that the sum of the angles of a triangle is greater than 180 degrees.)
It is a *very strong* critique of the "truth" of the conclusion to point out that the assumptions have not been proved.
The burden is on the proponent to prove the assumptions.
Zarquon: "Errol has posted a perfect example of the courtier's reply: you shouldn't diss Goedel or Platinga because hey, they're serious."
This is a perfect example of the use of the Courtier's Reply as a strawman argument. Errol was not saying "Don't diss Goedel or Plantinga," but rather saying that one shouldn't substitute sneer for rigorous argument.
John Lynch: "Ahem. Shouldn't you as a rationalist be disproving the assumptions and not just denying them. Whether Gödel's proof may be "destrpyed" by denying that "existance is a positive property", surely you must prove your stance rather than just deny Gödel's"
divalent: "That's *NOT* how it works. ... The burden is on the proponent to prove the assumptions."
The burden is on the claimant to prove the claim. Once you have made the claim that an assumption in a proof is faulty, you have to be able to defend it.
"Ironically, Errol has posted a perfect example of the courtier's reply: you shouldn't diss Goedel or Platinga because hey, they're serious. But theology, regardless of how seriously you take it is like Oakland: There's no there there."
Look, it is pointless to sit here and remotely insult one another. My comment about the seriousness of the position was meant to emphasize the fact that you cannot glibly dismiss well worked out arguments by philosophers the like of Gödel and Plantinga. I thought that some readers (certainly not you, because you already have it all figured out) might want to undertake an analysis of the ontological argument in a more intellectually honest way. Thus, I pointed them in the direction of some good books that do just that.
Finally, the majority of the commetors here might also be interested (or, if not interested, in need of) this website: www.fallacyfiles.org
JJ Ramsey: "The burden is on the claimant to prove the claim. Once you have made the claim that an assumption in a proof is faulty, you have to be able to defend it."
Good grief! So the truth of some conclusion is assumed until and unless the assumptions are proved false?
That is certainly not the way it is in math or science. Is this the way it is in philosphy and theology? [How does one disprove, say, that "Existance is an attribute of a perfect being?"]
It sound like a position that can only result in a pointless tennis match of "shifting the burden". How about:
- any proof of the existance of God based on an unproven assumption is invalid.
- all ontological arguments for the existance of God are based on unproven assumptions.
- Therefore, all ontological arguments asserting proof of the existance of God are invalid.
That should settle it
I'm going to have to get a copy of The God Delusion translated into Chinese, so I can wave it around and praise its irrefutability. Or maybe just its amazing depth.
Me: "Once you have made the claim that an assumption in a proof is faulty, you have to be able to defend it."
divalent: "Good grief! So the truth of some conclusion is assumed until and unless the assumptions are proved false?"
How you managed to get that non sequitur from my words is beyond me. What I said was clear: you have to be able to defend the claims that you make.
divalent: "How does one disprove, say, that 'Existance is an attribute of a perfect being?'"
Kant, I think, did a fair job.
PZ Myers: "I'm going to have to get a copy of The God Delusion translated into Chinese, so I can wave it around and praise its irrefutability. Or maybe just its amazing depth."
And your point is? No one suggested that expressing the ontological argument as modal logic made it irrefutable.
Me: "Axiom 5 in Godel's proof, 'Necessary existence is a positive property,' looks like it can be attacked by Kant's objection that "existence is not a predicate." Am I right?"
IEP: "necessary existence, unlike mere existence, seems clearly to be a property."
Ok, I guess I'm not right. I still see a problem, though. Not everything can be necessary. The statement 2+2=4 is necessary, but on the face of things, there is nothing that requires an omnimax entity to be necessary. I guess this mirrors Hume's argument that if one can conceive that something does not exist, then it is not necessary.
Right. And the axioms used to arrive at the logical proof are themselves claims, which have not been proven (else they wouldn't be axioms). It is these claims which hold the burden of proof, not the claim that they're specious. This is especially true in the ontological argument, which relies on all sorts of words with fuzzy meanings, like "perfect." Applying rules of logic to words with non-precise definitions is a sure-fire way to generate garbage.
In the context of generating garbage based on fuzzy language, I'll provide my favorite proof.
Theorem: All natural numbers are interesting.
Proof: Assume not. Then there must be a smallest natural number N which is not interesting. However, the property of being the smallest uninteresting natural number makes N interesting. Contradiction, therefore all natural numbers are interesting.
Why should anything be necessary? In your example, it is true as a group operation in integral domain of ordinary integers under arithmetic, and necessary for describing it, but it is not necessary true for other groups, with other applications in real life. Surely, nothing at all was an alternative to the somethings we see.
Davis: "Right. And the axioms used to arrive at the logical proof are themselves claims, which have not been proven (else they wouldn't be axioms). It is these claims which hold the burden of proof, not the claim that they're specious."
Not quite. If you merely observed that the axioms were not self-evident, the ball would still be in the court of those putting forth the ontological argument to show that the axioms were viable. However, once you go beyond saying that the axioms haven't been shown viable and make your own claim outright that they are specious, then you have to support it. An analogy may be helpful. If you simply note that someone has not substantiated their statements, you have no burden. If you claim that someone is lying or mistaken, you do.
Torbjörn Larsson: "Why should anything be necessary?"
I'm no expert in philosophy, but "necessary," AFAIK, here is a technical term meaning "must be true" or "must exist" or "has to be." "2+2=4" is necessary in the sense that it has to be true, so "2+2=5" is logically impossible. (Note: "2" is used to mean the integer "2", not a rounded representation of a number like 2.4.)
I think Torbjorn's point is that "2+2=4" is not necessary -- in fact, it's false in certain groups such as Z/(2Z) or Z/(3Z). In those, 2+2=4(mod 2)=0 and 2+2=4(mod 3)=1, respectively. 2+2=4 is only a necessary statement given the axioms of the real numbers (or more weakly, given the axioms of the natural numbers).
You're right -- saying they're "specious" is making the stronger claim that they're false, rather than the weaker one that they're unsubstantiated. Though I would be willing to go a little beyond that and say they're not well-defined.
It may be of interest to provide an English-like reading of the formal notation in the box. Here is my attempt -- note that I haven't the foggiest idea about _what_ system of modal logic is supposed to be used here, therefore all I'm doing is simply reading alound the notation. In the following, I use P and Q instead of phi and psi; iff means "if and only if"; P(x) and "x has property P" are synonimous.
Axiom 1. (Transitivity of positiveness) If it is necessary that for any x P(x) entails Q(x) then if P is a positive property Q is also a positive property.
Axiom 2. (Antisymmetry of positiveness) The opposite of a property P is positive iff P is not positive.
Theorem 1. (Contingent existence of positive objects) For any positive property P it is contingently the case that there is an x such that P(x).
Definition 1. (Divinity) We say that a propetry G is divine iff for any object x, G(x) means that x has all positive properties. (Note that therefore there is at most one divine property.)
Axiom 3. The divine property G positive.
Theorem 2. (Contingent existence of divine objects) Is is contingently the case that there is an object x which has the divine property G.
Definition 2. (Essentiality) We say that P is an essential property of an object x iff P(x) and for any Q such that Q(x) it is necessary that P(x) implies Q(x). (That is, P is an essential property of x if P(x) must hold for any Q(x).)
Axiom 4. (Self-necessity of positive properties) If a property P is positive then it is necessary that P is positive.
Theorem 3. (Essentiality of divinity) If an object x has the divine property G, then G is an essential property of x.
Definition 3. (Essence) We say that an object x is of the essence if for any property P which is essential of x it is necessary that there is an x such that P(x).
Axiom 5. (Positiveness of being of the essence) The property of being of the essence is positive.
Theorem 4. It is necessary that there is a divine object.
Note that Theorem 1 depends on what kind of modal logic is used. The rest of the derivations are straightforward, but they need 5 (five) axioms, some of which are less than self-evident.
There's a fundamental problem with this logic process. The reference shifts from something that is empiracally known, to something that is abstract. This is the same kind of shift that occurs in the common watchmaker analogy.
Finding a watch on the beach is evidence of a watchmaker. Therefore seeing a universe is evidence of a universe maker, and anything that can made a universe must be intelligent (god). This shifts from evidence for something that can be sensed and measured to something that cannot. The watchmaker and the god are very different types/classes. One is not abstract, one is.
Looks like a sophisticated version of the watchmaker to god analogy. A known relationship between known entities (watches are made by watchmakers) is used as the basis to declare unknown relationship and entities (the universe has a maker, AND the maker is god).
The "Courtier's Reply" criticism of people who reply to Dawkins is a strange one. The implicit assumption of the analogy is that one just needs to say "there's no there there" and point to theology, etc., and the whole thing will be unmasked and come tumbling down like a house of cards (as the Emperor's nakedness is revealed by the little boy's simply pointing out what all can see). But this analogy then raises the question why one would bother to write a several hundred page book arguing against a position that is supposed to be completely devoid of content.
I would like to know what criticism of Dawkins or reply to Dawkins would *not* be characterized as a "Courtier's Reply" by his defenders. It seems too easy a way to avoid argument. (And if argument is unnecessary, then why start one?)
Davis: "I think Torbjorn's point is that "2+2=4" is not necessary -- in fact, it's false in certain groups such as Z/(2Z) or Z/(3Z). In those, 2+2=4(mod 2)=0 and 2+2=4(mod 3)=1, respectively. 2+2=4 is only a necessary statement given the axioms of the real numbers (or more weakly, given the axioms of the natural numbers)."
If this is the point it's not a good one. The original claim was that 2+2=4, understood in the normal way as referring to the natural numbers 2 and 4 and the normal addition function on the natural numbers, is necessarily true. The fact that under a reinterpretation of the signs "2", "+" and "4," the string "2+2 = 4" can express a falsehood is both trivial and irrelevant. You might as well argue that 2 added to 2 could be 5, since we could have used the numeral "5" to refer to the number 4. Or that contradictions can be true, since we could have used the word "not" to mean "it is true that".
(By your reasoning, the proper response to Abraham Lincoln's question "If you called a tail a leg, how many legs would a donkey have?" would be "five." To which the correct response is, as Lincoln is supposed to have said: "No. Calling a tail a leg doesn't make it one.")
I think that Godel has proven in this that he knows exactly how many legs a donkey has by defining it in his terms.
No need for field research into whether 4 or 5 legged donkeys exist. The logic is sound and irrefutable.
Blast that Dawkins for demanding people produce a 5-legged donkey. What an infantile way to discuss this argument. After all, centuries of brilliant minds have wrestled with this question in many invented languages.
Sorry I haven't replied to the replies to may reply. Let me also apologize for the spelling mistake, I can see how that made all my points mute, and without merit.
I must admit that I find this claim rather strange:
"Ahem. Shouldn't you as a rationalist be disproving the assumptions and not just denying them. Whether Gödel's proof may be "destrpyed" by denying that "existance is a positive property", surely you must prove your stance rather than just deny Gödel's"
Perhaps I could have chosen my words more carefully, but my point is quite simply, that we can all invent premises and then show all we like.
The problem with the proof quoted above is that it does not define its premises clearly enough to show what they entail. What is a positive property? Is it sensible to define G(x) as it is done. The proof is incomplete in itself, perhaps not the logic, but if your premises and indeed your definitions are not supported, the proof itself is meaningless.
By denying the premises or definition I simply call for further clarification. Why is the definition of G(x) meaningful? As the Wikipedia entry says it leads to contradictions. Positive qualities can contradict each other, like mercy and justice, unmovable and adaptable etc.
What Dawkins does is introduce the layman to the fact that logical hoohay in itself is not an argument. You are free to examine the claims, even though the proof might be sound (which Gödel's doesn't seem to be, see the possible contradictions above)
And as other posters have remarked. The biggest problem with all these contrived proofs of gods existence, is ironically enough, the same thing that has been brought up against Dawkins book. They do not convince anyone of the existence of gods, just like people claim no one is converted to atheism after reading Dawkins.
At least Dawkins does not claim to have disproven the existence of gods, would it that Plantinga and all the other people toying with God proofs were so honest. To be fair, it would seem that Gödel did know that it wasn't a proof on par with his work in mathematics. I guess mathematicians are better suited than philosophers to being realistic ;)
"I guess this mirrors Hume's argument that if one can conceive that something does not exist, then it is not necessary."
Where does Hume argue this!? I would love to find out and read it...
Thanks!
You're implicitly saying it's necessarily true given the axioms of the natural numbers, which is what I said. This illustrates the problem of using natural language -- for someone familiar with the math, it's not unreasonable to ask what group/semigroup you're considering the operation to take place in if it's not stated explicitly (which it was not).
Davis; As I expect you well know, the "axioms of the natural numbers" (presumably the two-sorted first-order system known as Second Order Arithmetic) are themselves derivable in the following systems (not an exhaustive list)
1.First-Order Logic plus the ZF axioms;
2.First-Order Logic plus the NF axioms; &
3.Second-Order Logic plus the Hume-Cantor principle.
So if one believes that the axioms of one of 1,2 or 3 to be a priori necessary truths (a stretch, but just possible in the case of 3,. in my opinion) then surely 2+2=4 is then a necessary truth, _without_ taking "the axioms of the natural numbers" as a given?
I freely admit to not understanding Godel's argument
So you criticize Dawkins for not criticizing an obscure argument you don't even understand? Why bring it up in comparison, then? You don't even know what you're comparing, and would have know way of knowing how to compare them. "This makes no sense to me, refute it."
Adding on to PZ Meyer's comment, it's like saying Dawkins' arguments are "infantile" because he didn't deal with a Chinese translation of the Bible.
Christ, this is all some kind of joke, right? I mean, you can't seriously expect anybody to take this seriously? This is not serious. This isn't even... Fuck, I don't know what it is. It's just... What the Hell? No, really. What the Hell? What is going on here?
This is all too ridiculous. You folks are going to come together late and do a big reveal that this whole thing has been a joke, right? Is there a hidden camera somewhere? Is this an elaborate episode of PUNK'D?
Much as it pains me to agree with a guy calling himself "dorkarfork", that's the first thing that struck me about your post, as well. "Dawkins didn't take the work of refuting the Ontological Argument seriously - here, for example, is a a block of symbols I don't understand in the slightest and on whose actual meaning I won't comment." You might as well have just thrown up a couple photostats of the Dead Sea Scrolls.
The problems with Anselm's version lie partly in that it relies on word-trickery akin to the famous proof that "Ray Charles = God" by using loosely defined words like "great", partly that a reductio ad absurdum of the argument can be used to prove the existence of anything (e.g. "the unicorniest unicorn that can be conceived") but I've always felt the true weak point is in the assumption "one can conceive of a being greater than any other". You can't. You can put the words together, but ask yourself any simple questions about that being - "can it create a rock so heavy it can't lift it" et al - and you'll find you haven't actually got the ability to hold a conception of what a supreme being would be like in your head at all, any more than you can "conceive" of infinity. Godel bypasses that somewhat by expressing it in formal logic, the irony being that he's most famous for showing the limitations of formal systems, especially where infinite quantities are involved. The reductio still applies - take any physical quality and postulate an infinite amount of it, then point out that a thing that exists has more of that quality than a thing that doesn't and hey presto, you've proven the existence of the flyingest, spaghettiest, most monstrous Flying Spaghetti Monster of them all.
The fact that Dawkins largely breezes over and dismisses Anselm is not so much because he's seriously out to demolish the ontological proof as that the chapter was merely a quick survey of the "rational" arguments that have been put forward for the existence of god, all of which seem fairly unconvincing on their face, and none of which prove the existence of the *Christian* god. Those eager for more detail are presumably referred to the bibliography.
Again, the issue isn't Godel's "proof" per se. I have been consistently making the claim that Dawkins' chapter three bears little resemblance to philosophy of religion (contra theology) as actually practiced or for that matter has been practiced historically. Dawkins has taken "pop" theology and knocked it down. That's easy and, while it may satisfy the peanut gallery, it isn't really terribly interesting.
The chapters by Gale and Parsons in The Cambrdige Companion to Atheism cover much the same ground in a less dismissive fashion (and are very readable).
Hmm, I don't find the idea that mathematical axioms and logic are necessary truths convincing. We could just as easily assume some other completely different set of axioms; for example, I could imagine a strange culture where they took issue with an infinite set like the naturals, and instead insisted on only working in finite (but perhaps large) groups of the form Z/(pZ) for large primes p.
And what would it mean to say that First-Order or Second-Order Logic is necessary? That we couldn't create some different type of logic system?
Right. I have been meaning to blog about Gödel's proof myself, but I have not yet gotten around to it. A fellow statistician (that's not to say I'm in his league) has an analysis of Gödel's proof here.
In which case it would be titled:
Fà ng Pì
Davis: you interpret me as following:
"You're implicitly saying it's necessarily true given the axioms of the natural numbers, which is what I said. This illustrates the problem of using natural language -- for someone familiar with the math, it's not unreasonable to ask what group/semigroup you're considering the operation to take place in if it's not stated explicitly (which it was not)."
No, that's nuts. I'm quite familiar with the math. If I said that 2+2 was NOT 4, it'd be reasonable to ask me what group I was referring to, because clearly I couldn't be meaning by "2" etc what we all ordinarily mean by them. But if I say that 2+2 = 4, the default assumption is that I simply mean the number 2 and the number 4 and the ordinary operation of addition -- you know, the stuff you learned about in elementary school, before they confused you with all that mod 3 stuff.
Please note that I haven't said anything about axioms at all. People were adding numbers before there were any axioms for number theory.
Let me try to make the point another way. Suppose I grant you the right to ask me what group or semi-group I am referring you. So I tell you that I am talking about the ordered infinite group of the natural numbers, along with the addition operation on the natural numbers, and that by two I mean the successor of the successor of 0, and that by four I mean the successor of the successor of the successor of the successor of 0. So, that's what I mean by "2" and "+" and "4". So that's settled -- I'm not talking about addition mod 3 or equivalence classes of natural numbers, I'm talking about the natural numbers, in particular the natural numbers 2 and 4, and ordinary addition.
Now: is it possible that 2+2 is some number other than 4? Or isn't that a necessary truth?
"And what would it mean to say that First-Order or Second-Order Logic is necessary? That we couldn't create some different type of logic system?"
Well it is pretty difficult for me to see how one could seriously doubt the usual axiom schemata of those systems.
I suppose when I am feeling extra-skeptical, I am liable to some doubts about the law of the excluded middle. But even intuitionists get to use "2" and "4" don't they? Just not the class of natural numbers as a whole... Unless my memory of my few philosophy of maths classes is een more faulty than I usually suppose..
Davis: "Hmm, I don't find the idea that mathematical axioms and logic are necessary truths convincing. We could just as easily assume some other completely different set of axioms; for example, I could imagine a strange culture where they took issue with an infinite set like the naturals, and instead insisted on only working in finite (but perhaps large) groups of the form Z/(pZ) for large primes p.
And what would it mean to say that First-Order or Second-Order Logic is necessary? That we couldn't create some different type of logic system?"
No, that's missing the point entirely. The issue isn't about what logical systems we can create, but about what we can say once we've created a system (to put it in your terms, though of course our logical words had meaning before any logical systems were created). The question is whether, given the logical system that we have -- the system you rely on when you use words like "2" and "4" and "and" and "or" in ordinary practice -- and so the meanings that our words have, the sum of 2 and 2 could have been other than 4, or an axiom like if p then p, false.
Consider the following non-mathematical claims:
(1) Apples grow in Washington State.
(2) Apples are either apples or oranges.
The first claim is non-necessary (contingent) -- it could have been false. This is not because in some other culture the word "apples" might have meant bananas. It is because apples, those particular fruit that are crunchy, red, green or yellow, have little brown seeds in them, and so on, might never have been transplanted to Washington State. The possibility of a culture in which "apples" referred to bananas shows that the sentence -- in the sense of a string of inscriptions -- could have been false. But this doesn't show that what we say using that sentence could have been false -- that is a question about apples, not a question about words and how they are used.
The second claim is logically true, and so necessarily true. But again, notice that I didn't say that the second sentence could not have been false. I said that the second claim could not have been false. The claim, or proposition, is what we mean, express, by the sentence. Of course, there might have been a culture in which "or" was used to mean "and" -- in which the word "or" was governed by the logical principles which govern our word "and". So what? That shows only that the sentence "Apples are either apples or oranges" might have been used to say something false. It does not show that apples might have been neither apples nor oranges. Nor does it show that what we use the sentence "Apples are either apples or oranges" to mean might have been false. In fact, that couldn't possibly be false, and if we used the sentence "Apples are either apples or oranges" to say something false, that would show that we didn't mean by "or" what that word means in ordinary English.
It's more being a pedantic bastard than it is being nuts. I'm not actually being that much of a pedant, but I mostly wanted to point out that one could be, and would technically be correct.
The original claim, though, was that a certain set of axioms plus our standard logical systems are themselves necessary truths (unless I misunderstood). I'm not sure how one would even go about arguing that -- you're going to bump into Gödel if you try. I'm certainly not going to dispute that anything following from logic and our axioms is necessary, given those assumption.
"The original claim, though, was that a certain set of axioms plus our standard logical systems are themselves necessary truths."
I don't know about anyone else, but _my_ -somewhat contentious- claim is that the usual axiom schemata of 2OL express necessary truths. My - much more contentious - additional claim is that HCP (which is equivalent to a pretty strong Axiom of Infinity) is also a necessary truth. Additionally I claim that the rules of inference of 2OL preserve necessary truth. Hence - by Frege's theorem - all the usual 2OA theorems about the natural numbers are also necessary truths.
If you don't like axioms of infinity, replace HCP with the Finite Hume-Cantor principle and you will get a similar result (see Richard Heck's "Finitude and Frege's Theorem")
"you're going to bump into Gödel if you try"
I don't seem to have done.
I admit neo-logicism is not the most popular position in philosophy of mathematics, but I do maintain that it is tenable...
Anyhow if you don't like 2+2=4 as an example of a necessary truth, how about ~(P&~P)? Altho I gather there is current work in dialethic logics...
Does anyone who is less of a dilettante in these matters than I know whether any professional philosophers have argued that there are no necessary truths? It had honestly never ocurred to me before that there might not be... Thank you Davis, I think!
Yes, true in that system given these axioms. But not necessary to describe certain systems. Hence it is a model, contingent on application. The platonic view isn't necessarily correct. (Or "true". :-)
Spell-corrected version of my comment on the "Contemporary Discussions of the Ontological Argument" thread of the Mixing Memory science blog
http://scienceblogs.com/mixingmemory/2006/12/contemporary_discussions_o…
My frustration with the Millican paper was that it stopped right where I expected an interesting analysis of "Kripke's seminal 'Naming and Necessity' (1972) had made possible worlds and the occupants respectable...."
(p.472 in Mind, p.36 of 40 in the PDF)
That would also take us back to Vasiliev's invention of "Imaginary Logic" -- which has received recent metamathematical correction of its modern popularizer.
The treatment of God in terms of imaginary worlds, and/or fictional worlds, has to be done carefully. One can axiomatically and precisely discuss from our Universe #1 with Logic #1 the hypothetical occupants of Universe #2 with Logic #2, or, more subtly, Universe #2 with Logic #3 as alleged by someone in Universe #2 with Logic #2 about someone who it is claimed has a different logic. Proper language strips away the common errors of anthropologists and fiction critics alike.
An ontological argument in the proper Model Theory now becomes axiomatically possible. But, I fear, this has not been done by Kripke or Millican or anyone else I can find.
Posted by: Jonathan Vos Post |
December 10, 2006 03:03 PM