Proofs that God is imaginary

[Plotinus]

Two other concepts are buried in the definition, but perhaps not obviously so. Some of you have noted them. One is “Real” with a capital R and the other is “Everything” with a capital E. The use of capitals refers to an absolute rather than relative use of the word.

I'm still wondering what this part means. You say that God, as you believe him to be, is "Real" and "Everything", but in God's case these terms do not mean the same as the ordinary terms "real" or "everything". The difference is that in God's case they are absolute rather than relative. But what does this mean? There are of course well-known difficulties with the claim that terms mean something different when applied to God from what they mean normally, but in this case the difficulties seem greater. In what sense are the ordinary terms "real" and "everything" relative at all? To my mind, to say that something is real is not to make a relational statement at all; it is to attribute a property to the thing itself. For example, if I say that Obama is real but Gandalf is not real, I am not saying that Obama bears some kind of relation to something which Gandalf does not bear. I am saying that Obama has an intrinsic property which Gandalf lacks, the property of being actually instantiated. I don't see how that is a relational statement. If you think it is a relational statement, then what is the thing (or things) that Obama is related to that Gandalf isn't, in virtue of which the former is real and the latter is not? Furthermore, if (ordinary) reality is a relational matter, then if what God has is not relational, how is it similar to (ordinary) reality at all?

"Everything" is even harder, in my view, because this isn't even an adjective, but a sort of noun. Now some nouns can be relative, or at least express relations, such as "father". I suppose you could see "everything", in the ordinary sense, in this way. "Everything" is the thing or set that, for any given X, bears the relation of whole to part to X. But if that is what it means to say that "everything" is a relational term, what does it mean to say that God is "Everything" in a non-relational way?

Basically, what I'm saying here is that if you define these terms in a relational way, it is difficult to see what you mean by saying that God instantiates them in an absolute way. That's not to say that this can't be explained, but I'd like it more clearly set out!

I would find it difficult to hold both as true. They are very different types assumptions. My definition of god has utility in shaping a view of existence. Your ideas about the number 2 leads to expanding itself to an infinity of numbers and I do not see what that buys you in terms of making sense of things. Maybe you can build a model of existence from it, but I'm not sure I could. I can see how such a discussion would fit into a philosophy class, but it doesn't seem to me to answer any of life's questions. Assumptions that don't lead to where they can affect one's life, seem pretty silly. That is how I see the FSM and pink unicorn arguments.

I like this and it makes a lot of sense. But I'm not really sure it answers Fifty's point. As I understand it, Fifty is saying that your definition of God could equally well apply to other things, which are not God. If that's true then it must be at best insufficient as a definition and you need to add something which distinguishes God from those other things. Now you're saying that God, for you, shapes a view of existence, whereas the number two does not. But are saying that this is part of the definition of God? If you're not saying this, then you haven't answered Fifty, because it's still the case that your definition of God applies to things that aren't God. The fact that God has properties that those other things don't have (such as the property of shaping your view of existence) is neither here nor there as far as that problem goes, because if this property is not part of God's definition then it is a non-essential property, and numbers could still be God (on this definition) even without that property. So it seems that to address Fifty's problem (at least in this way) you need to say that shaping a view of existence is part of the definition of God. The problem with that is that it makes God's existence dependent on the existence of other people, because if other people didn't exist, God couldn't shape their existence. But depending upon the existence of other people does not seem compatible with the divine perfection.

It seems to me that even on a realist understanding of mathematical entities there are more useful ways one can distinguish between them and God. For example, numbers are abstract entities whereas God is a concrete particular.

Personally, my objection to your definition of God is that it doesn't seem to capture the notion of a perfect being. It seems to me that if God is anything he is a perfect being, such that a more perfect being not only does not exist but could not exist. He is, as they say, that than which no greater can be conceived. But your definition seems to me to leave out key perfections. I can imagine a being greater than your God, namely exactly the same thing but perfectly good, who always does what is best. I suppose you would respond to this by rejecting the definition of God as maximally perfect.

 

[Perfection]

Looks like the standard of debate here rose exponentially - good!

I'll fix that.

No perf you've got it backwards. God is Everything and the universe is the manifestation of the nothing that is latent within the Everything.

HAHA, you think you're a nothing.

 

[warpus]

My definition “points to” something that exists that is not bound by time or space or any physical dimensionality or physical properties, and that is permanent and never changes. It is all (Everything) that is Real (permanent and unchanging). Hence my frequent “God alone is,” posts. God is all that is Real. God is all encompassing in all dimensions, Everything with a capital E.

How do you reconsile the two bolded (bolding mine) ideas. Obviously stuff in the world changes, or I would never be able to type this.

Are you saying that the passage of time is an illusion and that the universe is at a stand-still, somehow?

 

[Flying Pig]

One of the main reasons that, I think, religion has grown up is that people like to think wishfully. They say (as one [I forget his name] bishop of some sort said in response to evolution) that science is a worldview which encourages selfishness and nihlism, that religion makes men good, etc etc. However,, that does not answer whether or not it is right!

Likewise,people spend their lives searching for a meaning of life when in fact all they are doing is running away from the idea that there is none.

 

[Phlegmak]

How do you reconsile the two bolded (bolding mine) ideas. Obviously stuff in the world changes, or I would never be able to type this.

Are you saying that the passage of time is an illusion and that the universe is at a stand-still, somehow?

BirdJaguar's definition of "god", as a sentient being, is simply a complete impossibility. The creature he describes cannot, and does not exist.

Also, if X = Everything, then why call it X? It simply needs no other name. "The universe" is a good enough name.

 

[Millman]

I love the tags. Imaginary or real I don't want to be controlled by it.

 

[Birdjaguar]

My mother in law had a stroke today; I will not be posting tonight.

 

[Fifty]

My mother in law had a stroke today; I will not be posting tonight.

I hope everything works out as best as it can!

 

[Millman]

I hope your mother gets all right. And I say this willingly because I think everyone hates being hurt.

 

[Atticus]

Sorry to hear about your ma in law, Birdjaguar, hope she gets better.

For instance, the number 2.

But number 2 is not infinite

More seriously, passages like this:

My definition “points to” something that exists that is not bound by time or space or any physical dimensionality or physical properties, and that is permanent and never changes. It is all (Everything) that is Real (permanent and unchanging). Hence my frequent “God alone is,” posts. God is all that is Real.

gives the idea that Birdjaguar doesn't believe there are multiple gods. Therefore his definition is to be understood to mean that there is one god, which consists of everything that satisfies these and these conditions. Number 2 wouldn't then be god, because there are other things (number 3 and so on), which satisfy the conditions, they would be only part of god. Therefore counter examples aren't probably good to show his definition bad.

You could of course find something that you think satisfies the conditions, but is so vile that can't be called god, but even that wouldn't do the job, since it would be only part of the god. For example, you could say that being put into jail sucks, but when thought from the society's view it's good thing. Or pain sucks, but from the view of organism, it's a good thing too. The only way to produce counter example would therefore be to make list of everything which satisfies the conditions, and to show that they taken together don't deserve to be called god.

And of course I believe that Birdjaguar is mystic, so he never was going to give the accurate description, but rahter a starting point for speculation. This I said only to point out that by speculating our purpose isn't and shouldn't be to put him in front of an inquisition.

"Eternal" is traditionally taken to mean either infinitely old, or outside time altogether, so neither old nor new.

Oh boy, pretty amazing that I've lived this old without knowing that...

Whether or not this proof works, I'm not sure. Leibniz thought that the ontological argument would work if you could show that God is possible, which Descartes never did; he thought that by supplying the proof outlined above he had made good the deficiency and that the ontological argument could now be defended as a deductive argument for God's existence. I'm pretty sure he was wrong about that part.

Kurt Gödel wrote a proof based on Leibnitz' proof using modal logic. You can find it in the third volume of his collected works, but it's in very brief form there. However there's some articles about it which go more into detail. Gödel himself left the proof unpublished because he didn't want anyone to think he believes in god.

 

[Fifty]

Kurt Gödel wrote a proof based on Leibnitz' proof using modal logic.

Of course, its quite easy to see that if the Anselmian conception of God is possible, then it is actual, when we use modal logic. The greatest way to exist is to necessarily exist, so if one of God's properties is necessary existence, then if he's possible then he's actual, based on elementary possible world semantics. Does Gödel extensively address the question of whether such a God is possible? Because that's the interesting part.

 

[Atticus]

Yeah, he proves it. It's been some years since I went through the proof, so I don't remember it that well, but I'll look into it when I go to the library.

Here's how I remember it went:

notation:
A(x) means that x has property A
Pos(A) means that property A is positive, what is positive property is specified in the axioms later
I(q) means that proposition q is impossible
P(q) means that proposition q is possible
=>, <=>, & are normal connectives, - stands for negation.
(x) means "for all x in Omega", where the Omega is the universe, or basic set, or wahtever it is called.

|=> is entailment, also known as the strict implication. Definition of p|=>q is that it's impossible that p is true and q not: I( p& -q ). The difference with normal, material implication is that p=>q doesn't mean it's impossible that p&-q, it can be just a matter of fact.

Axioms:
Axiom 1. Pos(A) & (x)( A(x) |=> B(x)) => Pos(B)
If property B is entailed by property A, and A is positive, then also B is positive

Axiom 2. Pos(-A) <=> -Pos(A)
If property -A is positivie, then A isn't, and vice versa.

Definition 3. x is God-like, if (A) ( Pos(A) <=> A(x)). That is, x is God-like, if it has all the positive properties and only them. If x is God-like, we'll write G(x).

Axiom 4. Pos(G).

There are other axioms too, but I'm going here just for the possibility proof for which they are irrelevant.

Lemma 5. If it's impossible that p, then p |=> q for any proposition q.

proof: Choose arbitrary proposition q. If p is impossible, then p&r is also impossible for any r. Choose r=-q, and you get I(p&-q), which is what we wanted to prove.

Theorem 6. If A is positive property, then it's there is x for which A(x) is possible.

proof. Suppose that A is positive property, and for every x it's impossible that A(x). So if we choose x, lemma 5 tells us that A(x) entails any proposition, for example -A(x). As a result we have
(x)( A(x) |=> -A(x)),
and because A is positive, it follows from axiom 1 that also -A is positive. This however is contradiction with axiom2, so we have proved the claim.

Corollary 7. It is possible that there is God-like being.

proof: result from Theorem 6 and axiom 4.

Theorem 8. It is possible that I remember the proof wrong.

 

[Plotinus]

My mother in law had a stroke today; I will not be posting tonight.

Sorry to hear that, Birdjaguar. I hope she's OK.

Of course, its quite easy to see that if the Anselmian conception of God is possible, then it is actual, when we use modal logic. The greatest way to exist is to necessarily exist, so if one of God's properties is necessary existence, then if he's possible then he's actual, based on elementary possible world semantics.

Woah! Wait a moment. You're saying you think the ontological argument works? I'd like to see this in more detail.

Yeah, he proves it. It's been some years since I went through the proof, so I don't remember it that well, but I'll look into it when I go to the library.

Here's how I remember it went:

notation:
A(x) means that x has property A
Pos(A) means that property A is positive, what is positive property is specified in the axioms later
I(q) means that proposition q is impossible
P(q) means that proposition q is possible
=>, <=>, & are normal connectives, - stands for negation.
(x) means "for all x in Omega", where the Omega is the universe, or basic set, or wahtever it is called.

|=> is entailment, also known as the strict implication. Definition of p|=>q is that it's impossible that p is true and q not: I( p& -q ). The difference with normal, material implication is that p=>q doesn't mean it's necessary that p&-q, it can be just a matter of fact.

Axioms:
Axiom 1. Pos(A) & (x)( A(x) |=> B(x)) => Pos(B)
If property B is entailed by property A, and A is positive, then also B is positive

Axiom 2. Pos(-A) <=> -Pos(A)
If property -A is positivie, then A isn't, and vice versa.

Definition 3. x is God-like, if (A) ( Pos(A) <=> A(x)). That is, x is God-like, if it has all the positive properties and only them. If x is God-like, we'll write G(x).

Axiom 4. Pos(G).

There are other axioms too, but I'm going here just for the possibility proof for which they are irrelevant.

Lemma 5. If it's impossible that p, then p |=> q for any proposition q.

proof: Choose arbitrary proposition q. If p is impossible, then p&r is also impossible for any r. Choose r=-q, and you get I(p&-q), which is what we wanted to prove.

Theorem 6. If A is positive property, then it's there is x for which A(x) is possible.

proof. Suppose that A is positive property, and for every x it's impossible that A(x). So if we choose x, lemma 5 tells us that A(x) entails any proposition, for example -A(x). As a result we have
(x)( A(x) |=> -A(x)),
and because A is positive, it follows from axiom 1 that also -A is positive. This however is contradiction with axiom2, so we have proved the claim.

Corollary 7. It is possible that there is God-like being.

proof: result from Theorem 6 and axiom 4.

Theorem 8. It is possible that I remember the proof wrong.

This is very interesting. Thanks for posting it. And if that's all from memory then I'm very impressed. But I'm not sure this proves that a being with all the positive qualities is possible. As far as I can see, it purports to prove that each positive quality, individually, is possible, that is, it is possible that each one is instantiated. That is what Theorem 6 proves. But I don't see that it proves that it is possible that one and the same individual instantiates all of them. It does that only by combining Theorem 6 with Axiom 4. But surely what's required is a proof of Axiom 4 - isn't it?

In other words, I think there may be some confusion here, of the kind that slips from "Every positive quality may be instantiated by some entity" to "There may be an entity which instantiates every quality". Like the old joke about "A woman gives birth every two seconds - she must be exhausted."

 

[Fifty]

Woah! Wait a moment. You're saying you think the ontological argument works? I'd like to see this in more detail.

I'm not saying it proves that God exists, just that it proves that if one of God's properties is necessary existence, and if God possibly exists, then God exists.

I don't want to write it out in modal logic notation, but its easy enough to see if you just frame things in terms of standard possible world talk. Something that necessarily exists exists in all possible worlds. Something that possibly exists exists in at least one possible world. If God possibly exists, then he exists in at least one possible world. And if he exists in one possible world, then he exists in every possible world, because thats just what it means to necessarily exist.

notation:
A(x) means that x has property A
Pos(A) means that property A is positive, what is positive property is specified in the axioms later
I(q) means that proposition q is impossible
P(q) means that proposition q is possible
=>, <=>, & are normal connectives, - stands for negation.
(x) means "for all x in Omega", where the Omega is the universe, or basic set, or wahtever it is called.

|=> is entailment, also known as the strict implication. Definition of p|=>q is that it's impossible that p is true and q not: I( p& -q ). The difference with normal, material implication is that p=>q doesn't mean it's necessary that p&-q, it can be just a matter of fact.

Axioms:
Axiom 1. Pos(A) & (x)( A(x) |=> B(x)) => Pos(B)
If property B is entailed by property A, and A is positive, then also B is positive

Axiom 2. Pos(-A) <=> -Pos(A)
If property -A is positivie, then A isn't, and vice versa.

Definition 3. x is God-like, if (A) ( Pos(A) <=> A(x)). That is, x is God-like, if it has all the positive properties and only them. If x is God-like, we'll write G(x).

Axiom 4. Pos(G).

There are other axioms too, but I'm going here just for the possibility proof for which they are irrelevant.

Lemma 5. If it's impossible that p, then p |=> q for any proposition q.

proof: Choose arbitrary proposition q. If p is impossible, then p&r is also impossible for any r. Choose r=-q, and you get I(p&-q), which is what we wanted to prove.

Theorem 6. If A is positive property, then it's there is x for which A(x) is possible.

proof. Suppose that A is positive property, and for every x it's impossible that A(x). So if we choose x, lemma 5 tells us that A(x) entails any proposition, for example -A(x). As a result we have
(x)( A(x) |=> -A(x)),
and because A is positive, it follows from axiom 1 that also -A is positive. This however is contradiction with axiom2, so we have proved the claim.

Corollary 7. It is possible that there is God-like being.

proof: result from Theorem 6 and axiom 4.

This is extremely confusing. I tried clarifying it by writing it down on paper and making it as formal as possible, but my confusion remains. The proof of Thm. 6 is weird, and might violate some of the standard rules of quantifier logic (though this is based on just a glance)... of course that wouldn't be a problem if the logistic system defined were defined more rigorously. Also, I don't see how the denial of your reductio assumption in the proof of thm. 6 is equivalent (in the system you defined) to the theorem to be proved. There are various other points of confusion, but its not wroth going into right now. Of course, I'm not blaming you or anything, just saying that not much can be said about this as-is... let us know when you've had a chance to check out the full proof again (perhaps a citation so I can see if its in my library and check it out for myself?)

 

[Atticus]

Here's one little typo of mine corrected:

The difference with normal, material implication is that p=>q doesn't mean it's impossible that p&-q, it can be just a matter of fact.

And if that's all from memory then I'm very impressed. But I'm not sure this proves that a being with all the positive qualities is possible. As far as I can see, it purports to prove that each positive quality, individually, is possible, that is, it is possible that each one is instantiated. That is what Theorem 6 proves. But I don't see that it proves that it is possible that one and the same individual instantiates all of them.

It's not that impressive really, all you have to do is to remember the idea and fill in the rest. I've studied mathematics, so I'm used to proving things.

You're right that thm 6 doesn't prove that it's possible for one thing to have all the positive properties. But since G is positive, you can apply it only to this one property.

Axioms don't need to be proven, but of course you don't have to accept the conclusion if you don't accept the axioms. However axiom 4 isn't very problematic: If having all the positive properties isn't positive property, then what is?

Gödel might have used instead an axiom that arbitrary collection of positive properties is positive, from which our axiom follows. It's perhaps better formulation, because it shows that the properties should be simple, like you said on previous page.

let us know when you've had a chance to check out the full proof again (perhaps a citation so I can see if its in my library and check it out for myself?)

It was in Kurt Gödel's "collected works", volume 3. Edited by Solomon Feferman, and it has it's own heading, so once you find the book, the proof is easy to find. It's pertty briefly there, but there's couple of articles about it too, which I'll try to find also.

Unfortunately I'm not friend of formal proofs, but I'll try to explain theorem 6 better, if someone else is interested too.

Theorem 6. If A is positive property, then for some x it's possible that A(x).

proof. We use reductio ad absurdum: Let us suppose that the claim isn't true, for some positive property A there is no such x that A(X) is possible.

Now pick an arbitrary x. Since A(x) is impossible, any proposition is entailed by it, especially -A(x). We get:
A(x) |=> -A(x).

Now since x was arbitrary the above holds for every x:
(x) (A(x) |=> -A(x) ).

Because A is positive property, it follows from this and axiom 1 that -A is a positive property also. But axiom 2 forbids both A and -A being positive, so we have arrived at contradiction. Therefore the supposition that thm 6 doesn't hold proved out to be wrong.

 

[Plotinus]

I'm not saying it proves that God exists, just that it proves that if one of God's properties is necessary existence, and if God possibly exists, then God exists.

Ah, but you said that you can get to this from the Anselmian definition of God, not merely the claim that God has the property of necessary existence. That's not an Anselmian property. Now one could argue that it follows from the Anselmian definition, using the argument you suggested before - to exist necessarily is the greatest way of existing, so that than which no greater can be conceived must exist necessarily (if it exists). But I'm not at all convinced that necessary existence is "greater" than ordinary existence. Anselm himself, of course, doesn't argue in this way.

I don't want to write it out in modal logic notation, but its easy enough to see if you just frame things in terms of standard possible world talk. Something that necessarily exists exists in all possible worlds. Something that possibly exists exists in at least one possible world. If God possibly exists, then he exists in at least one possible world. And if he exists in one possible world, then he exists in every possible world, because thats just what it means to necessarily exist.

I'm not convinced that this argument works, even if we accept the possibility of a necessarily existing object. The standard objection to it is a reductio - if this argument works then we can define all sorts of things as necessarily existing, including a being such that no maximally excellent being co-exists with it in the same possible world. You can construct an ontological proof of God's non-existence in this fashion:

(1) The property of no-maximality is such that if a being exemplifes this property in possible world A, there can be no maximally excellent being in possible world A.
(2) The property of superlative no-maximality is such that a being with this property exists in every possible world, and has the property of no-maximality in every possible world.
(3) The property of superlative no-maximality is possible (ie, there is a possible being that exemplifies it).
(4) So there is a possible world containing a being of superlative no-maximality.
(5) But if such a being exists in one possible world, it exists in every possible world.
(6) Therefore, a being actually exists with the property of no-maximality.
(7) Therefore, no being with the property of maximal excellence actually exists.

(I forget who originally formed the argument in this way.)

But this sort of thing seems absurd, in which case there must be something wrong with the original. It seems to me that (5) is the problematic part of this and its counterparts. The mere fact that a being that exists in every possible world is possible may entail that it exists in some possible world; but it doesn't follow from this that it exists in every possible world. It follows only that it exists in every possible world in that possible world. In other words, a possible necessity is not the same thing as necessity. Considerations of this sort have led some people to suppose that the standard Plantinga-style modal logic that this form of argument depends upon is flawed, and that possible worlds may be considered to "nest" within one another rather than all be on the same modal footing, as it were.

 

[HannibalBarka]

1. why would proving or disproving the existence of A God so important?
If the purpose was to prove or disprove the existence of the God of the Bible of the Koran or some other "real" God (ie some God that is worshipped by some one), than I can see the utility of this quest and its impact at least on those people who worship or consider the possiblity of worshipping him; but proving the existence of a "theorical God" isn't of any practical use, is it? Disproving the possiblity of existence of ANY GOD maybe, as it would negate the existence of those particular gods described above, but isn't this some non-achiavable quest (I mean we have a hard time disproving the non exitence of some specific Gods already)?

2. do you feel there is a true positive evolution in Philisophy as there is in science for example? tis quest about God for example, do you think we have acheived much progress about it compared to where Platon was? the same way we progressed concerning Astronomy to where Euclide was? isn't Philosophical progress closer to the kind of religious progress (ie we're just accumulating new ways of explaining things but not necessarily better ones) than to scientific progress (where new theory is better than old one)?

 

[Plotinus]

1. why would proving or disproving the existence of A God so important?
If the purpose was to prove or disprove the existence of the God of the Bible of the Koran or some other "real" God (ie some God that is worshipped by some one), than I can see the utility of this quest and its impact at least on those people who worship or consider the possiblity of worshipping him; but proving the existence of a "theorical God" isn't of any practical use, is it? Disproving the possiblity of existence of ANY GOD maybe, as it would negate the existence of those particular gods described above, but isn't this some non-achiavable quest (I mean we have a hard time disproving the non exitence of some specific Gods already)?

Does it matter whether it's of practical use, though? Isn't it of at least some theoretical interest whether the universe was created by a divine being or not, even it makes no practical difference?

2. do you feel there is a true positive evolution in Philisophy as there is in science for example? tis quest about God for example, do you think we have acheived much progress about it compared to where Platon was? the same way we progressed concerning Astronomy to where Euclide was? isn't Philosophical progress closer to the kind of religious progress (ie we're just accumulating new ways of explaining things but not necessarily better ones) than to scientific progress (where new theory is better than old one)?

This isn't really on-topic, but still. It's often said that one of the key differences between philosophy and science is that old science becomes obsolete as the subject advances, whereas old philosophy doesn't. I'm not really convinced that that's true. We know things in philosophy today that we didn't know in the 1960s, for example, such as that emotivism is an inadequate meta-ethical theory. And philosophy has certainly come a long way since Plato (whatever Whitehead may have thought): the modal logic used in the preceding posts was not developed until the late Middle Ages at the earliest. The difference between philosophy and science that unequivocal evidence for or against particular positions in philosophy tends not to be forthcoming while it is in science. This means that no philosophical position can ever be made completely obsolete in the way that a scientific one can. But this is just part and parcel of philosophy. Philosophy deals with things that can't be settled by what we can narrowly call "the scientific method". Or, more accurately, "science" is that part of philosophy that deals with things that can be settled in this way. When we consider matters that can't be settled by the scientific method, such as the existence of God, we naturally lose the resources of that method and the kind of certainty it can provide. That's not philosophy's fault, though - philosophy is the attempt to consider these matters as critically and carefully as we can even where it is impossible to apply the scientific method.

 

[Flying Pig]

Does it matter whether it's of practical use, though? Isn't it of at least some theoretical interest whether the universe was created by a divine being or not, even it makes no practical difference?

Welcome to philosophy

 

[Atticus]

I took a look at the proof in Gödel's collected works, and it was even more cryptic there than I recalled, Gödel wasn't friend of formal proofs either. However there's a foreword by Robert Adams (starting at page 388), which clarifies the proof a bit and puts it into context. He also gives some references to other articles, and talks about Leibnitz quite a bit too (I refuse to write it without t).

Also very recommended reading is Dana Scott's and Jordan Sobel's articles in
On Being and Saying: essays for Richard Cartwright ed. Thomson, Judith Jarvis, Cambridge, Mass. MIT Press 1987.
They help quite a bit understanding the proof, contain copies of the original, and latter also critcism. I should have copies of them somewhere, but can't find them.